Designed monoclonal antibodies are attracting growing interest for their therapeutic potential for viral and bacterial infections, cancer, autoimmune diseases, infertility, and even contraception. Currently most monoclonals are raised in animals and/or refined by in vitro affinity maturation. In any event this process is too slow to be useful as a reaction to viral disease outbreaks and often results in a product that is immunogenic, of the wrong binding affinity, or which binds the wrong epitope. There is therefore a demand for an in silico affinity maturation method which can quickly and reliably generate monoclonals of a given affinity to a specific target. Previous attempts have used rotamers to economically optimize side chain conformations, but have a fatal flaw: they do not properly take into account backbone motions and so often fail to detect mutations which would result in improved binding. Molecular Dynamics in principle could solve this problem, but because there are so many degrees of freedom in an antibody-antigen complex, and many possible mutants to evaluate, this is simply not economical. We present a solution to this problem: a fast equilibration in torsion angles near the site of the mutation, followed by more accurate evaluation of binding energy. Using this method, we accurately and economically recapitulate the experimentally measured binding affinities of the monoclonal D44.1 against chicken egg white lysozyme. The results suggest a wide variety of future applications.