HYSTERESIS AND STATISTICAL ERRORS IN FREE-ENERGY PERTURBATION-L TO D-AMINO-ACID CONVERSION

A theory based on a Langevin equation along the reaction coordinate is developed to explain and calculate systematic and statistical errors in free energy perturbation simulations. The errors are calculated exa ctly when both the perturbation potential and the mean potential from the surrounding degrees of freedom are harmonic in the reaction coordi nate. The effect of the mean potential is small as long as the force c onstant is small compared to the force constant of the perturbation po tential. This indicates that the results obtained with zero mean force may still be valid as long as the second derivate of the mean potenti al is small compared to that of the perturbation potential. The theory is applied to conversion between L and D amino acids by changing the position of the minimum of the harmonic improper dihedral potential be tween +/-35.264 degrees. For phenylalanine bound in the active site of a protein (thermolysin) we find from 20 psec. simulations statistical errors and hysteresis that both are about 2.5 kJ/mol in agreement wit h what is obtained from the theoretical predictions. The statistical e rrors are proportional to the square root of the coupling to the heat bath and inversely proportional to the square root of integration time while the (positive) hysteresis due to that the reaction coordinate l ags behind is linear in the same quantities. This shows that the syste matic errors will dominate in short simulations while the statistical ones will dominate for long simulations. The treatment is based on tha t the systematic influence of the surroundings can be represented by a mean force upon the reaction coordinate. If the relaxation processes of the environment are slow this may not be true. Then additional erro rs have to be considered.