Strategies to elicit Abs that can neutralize diverse strains of a highly mutable pathogen are likely to result in a potent
vaccine. Broadly neutralizing Abs (bnAbs) against HIV have been isolated from patients, proving that the human immune system
can evolve them. Using computer simulations and theory, we study immunization with diverse mixtures of variant antigens (Ags).
Our results show that particular choices for the number of variant Ags and the mutational distances separating them maximize
the probability of inducing bnAbs. The variant Ags represent potentially conflicting selection forces that can frustrate the
Darwinian evolutionary process of affinity maturation. An intermediate level of frustration maximizes the chance of evolving
bnAbs. A simple model makes vivid the origin of this principle of optimal frustration. Our results, combined with past studies,
suggest that an appropriately chosen permutation of immunization with an optimally designed mixture (using the principles
that we describe) and sequential immunization with variant Ags that are separated by relatively large mutational distances
may best promote the evolution of bnAbs.