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11.5.1 Theory for NOE Constraints

The NOE constraint function is designed improve upon a flat bottomed harmonic constraint function. The problem with the harmonic constraints is the high forces placed on atom pairs that are very far from their constraint distance. The potential in CONGEN uses a harmonic potential if the distance is close to correct, or if the distance is less than the lower bound of the constraint. However, at large distance, the constraint force is constant rather than harmonic. The connection between the harmonic section of the potential and the constant force section is done by an inverted harmonic piece. The main advantage of this functional form is that a user can input all constraints at once in the beginning of the run.

The functional form is as follows:

     ENOE  = sum_over constraints
              0.0                                  if  r_l < r_p_k < r_u
              K_i * (r_p - r_l) ^2                 if  r_p < r_l
              K_i * (r_u - r_p) ^2                 if  r_u < r_p < r_sw
              -K_i * (r_plat - r_p) ^2 + (2*Emax)  if  r_sw < r_p < r_out
              K_i * SLOPE * (r_out r r_p) + Emax2  if  r_p > r_out


     i is taken over all NOE's,
     K_i is the weight associated with the constraint,
     r_eff is the effective interproton distance calculated from a single structure
     or over an ensemble of structures (see below),
     r_l is the lower bound of the constraint,
     r_u is the upper bound of the constraint,
     r_sw is the point where the function switches to an inverted harmonic, and
     has the form, r_sw = FMAX / 2 K + r_u,
     r_plat is the point where the derivative of the inverted harmonic would go to
     zero and has the form, r_plat = 2 r_sw - r_u,
     r_out is the point where the function goes linear and has the form,
     r_out = r_plat - SLOPE / 2 K_i,
     Emax is the energy where the inverted harmonic switches in and is
     equal to K ( r_sw - r_u ) ^ 2, and
     Emax2 is the energy where the function goes linear and has the form,
     Emax2 = 2 * Emax - K ( r_out - r_plat ) ^ 2.

The form of this function is given in the following figure where the lower bound is 2 Angstroms, the upper bound is 4 Angstroms, K = 2, SLOPE = 0.5, and FMAX = 4.0. Values for the various distances were computed from these parameters.

                              NOE CONSTRAINT FUNCTION
               8. ***********************************************
                  **                                            *
                  **                                            *
               6. **                                           **
           E      * *                                           *
           N      * **                                        ***
           E      *  *                                 ******** *
           R    . ** **                           ******       ***EMAX2
           G      *   *                          **             *
           Y      *   **                        **              *
                  *    **                      **               *
               2. **    **                    **               **EMAX
                  *      **                  **                 *
                  *       **               ***                  *
               0. **********R0L*********R0U**R0SWR0OUT***********
                 0.0        2.0         4.0  5.0 5.875         8.0
                                 EFFECTIVE DISTANCE

NOE constraints can be used in both standard and ensemble-averaged calculations. In a standard calculation, r_eff is just the distance derived from the individual structure being refined. In the ensemble average approach, multiple different structures, currently stored as different segments separated in space, are refined simultaneously, with distances for NOE constraints being computed using the following formula:

     r_eff = (1/n sum k from 1 to n (r_p_k ^ -x) ) ^ (-1/x)
     r_eff is the effective interproton distance used in E_noe below,
     k ranges over the conformations in the ensemble,
     n is the number of conformations in the ensemble for which r_p_k can
     be calculated,
     r_p_k is the interproton distance as calculated below, and
     x is 6 or 3, depending on the averaging conditions.

Each member of the ensemble must have identical atoms and residues as determined by their IUPAC names and residue identifiers.

In cases where the NOE is due to the interaction of motionally averaged or prochiral protons, the interproton distance, r_p, is calculated over all possible pairing of two sets of protons. By default, the following expression is used:

r_p = ( sum (1/ri)^6) ) ^ (-1/6)

where the sum is taken over all possible pairs of atoms involved in the constraint. The constraints are specified using two sets of atom selections, see Atom Selection, so that any combination of atoms may be specified.

If NOE intensity scaling is done such that the calculated distances should reflect averages instead of sums, the AVERAGE option can be used when specifying the constraint. In this case, the following expression is used:

r_p = ( (sum (1/ri)^6) / N) ^ (-1/6) where N is the number of pairs used in the average. If any atom in the two sets of atoms in the constraint has undefined coordinates, then the interproton distance is omitted from other calculations. Likewise, if all the atoms in the two sets are fixed, see Fixed Atoms, the distance is ignored.