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11.5.2 Theory for J Coupling Constraints

The J coupling constraint term allows one to incorporate constraints based on scalar J coupling measurements in two different ways, one where all atoms in the J coupling are known, and the other where two measurements are made each involving one proton in a pro-chiral pair, but for which the prochiral assignment is unknown. In this section, the form of the scalar J coupling equation will be discussed first followed by the two forms of the constraint equations.

The J-coupling constraint is based on the Karplus equation.12

     J = c_1 cos^2(phi) + c_2 cos(phi) + c_3

where J is the calculated value for the coupling constant given the current coordinates of the system.

phi is a torsion angle defined over four bonded atoms.

c_1, c_2, c_3 are coefficients for the coupling constant. The default setting for these coefficients is 6.4, -1.4, and 1.9 (the value for the J_HNHa), but they can be changed for any constraint.

Ensemble averaging can be performed over a set of constraints as long as each constraint belongs to a different segment with the same structure, as described for the NOE ensemble averaging, see Theory for NOE Constraints. In the case of averaging, J is computed as follows:

     J = c_1 <cos^2(phi)> - c_2 <cos(phi)> + c_3

where the averages are computed over the members of the ensemble.

There are two ways to incorporate constraints from these scalar coupling constants into the energy function. When all the atoms involved in a scalar coupling constant are known, the following equation can be used:

                { 0.0                    if J_l <= J <= J_u }
     EJCP = sum { K_j K_n (J - J_l) ^ 2  if J < J_l         }
                { K_j K_n (J - J_u) ^ 2  if J_u < J         }

where i ranges over all J coupling constraints. Every parameter in the equation for E_j can be varied for each constraint.

J_l and J_u are the lower and upper bounds for a measured J coupling constant.

J is the calculated value for the coupling constant given the current coordinates of the system.

When one of the atoms involved in the J coupling constant measurement is a prochiral atom and if coupling constants for both prochiral atoms are known but unassigned, then the following form of the equation can be used. In this equation the two J coupling constants are “joined” together, and the constraint function is computed based on relationships involving the sums and magnitudes of the differences of the two J couplings, which obviates the need for stereospecific assignments. In the constraint file, the JOIN command is used to link to measured J constraints together.

In this functional form, the sum term is just a harmonic restraint based on the sum. The difference term is more complex because the sign of the difference depends on the arbitrary choice chirality on the prochiral center. The difference function V_d is harmonic where the magnitude of the calculated difference is bigger than the experimental difference. If the magnitude of the calculated difference is less, then a piecewise harmonic function with a maximum at |J_1 - J_2| = 0 is used. A additional factor is used for this term to smooth the overall function.

E_j = K_j K_n (V_s + V_d)

           { ((J_1 + J_2) - JS_max)^2  if |J_1 + J_2| > JS_max            }
     V_s = { 0                         if JS_max >= |J_1 + J_2| >= JS_min }
           { (JS_min - (J_1 + J_2))^2  if |J_1 + J_2| < JS_min            }
           { (|J_1 - J_2| - JD_max)^2         if |J_1 - J_2| > JD_max             }
     V_d = { 0                                if JD_max >= |J_1 - J_2| >= JD_min  }
           { k'(JD_min - |J_1 - J_2|)^2       if JD_min/2 <= |J_1 - J_2| < JD_min }
           { k'(JD_min^2 / 2 - |J_1 - J_2|^2) if 0 <= |J_1 - J_2| < JD_min/2      }

JS_max = Max(|JS + eps|, |JS - eps|)

JS_min = Min(|JS + eps|, |JS - eps|)

JD_max = Max(|JD + eps|, |JD - eps|)

JD_min = Min(|JD + eps|, |JD - eps|)

eps = sqrt(eps_1^2 + eps_2^2)

JS = Jexp_1 + Jexp_2

JD = Jexp_1 - Jexp_2


K_n is a optional normalization factor. If it is used, it is set to 1 / Max(E_j) for E_j calculated as described for the JNORM option below.

J_1 and J_2 are the calculated values for the two scalar coupling constants involved the two prochiral atoms.

Jexp_1 and Jexp_2 are the experimental values for the scalar coupling constants. These are calculated from the average of the lower and upper bounds specified in the constraint file for the J coupling.

eps_1 and eps_2 are calculated from the experimental scalar coupling as half the difference of the upper and lower bounds on the coupling constants.

k' is a scale factor to adjust the difference term down to eliminate local minima in E_j.


[1] M. Karplus, J. Chem. Phys. 30, 11-15 (1959).

[2] M. Karplus, J. Am. Chem. Soc. 85, 2870-2871 (1963).