Optimal Maturation Protocols for High-Affinity Antibody Targets: A Path-Integral Approach

Affinity maturation is a stochastic evolutionary process allowing the adaptive immune system to produce B-cells capable of recognizing antigenic molecules. One of the main factors influencing the quality of the maturation outcome, quantified by the affinity of the produced antibodies to the antigen, is the time-course of antigen availability. In this paper, we introduce a stochastic model for affinity maturation, calibrated against in vivo lineage dynamics and deep mutational scanning data, and address the following question: what is the best antigen-concentration protocol maximizing the probability to reach a target affinity value at the end of the process? We introduce a path-integral formalism to identify the maturation trajectories of the rare clones ending at the desired target affinity, estimate their probabilities, and maximize them over the antigen-concentration temporal profile. The theoretical optimal concentration protocol is approximated by a discrete three-injection schedule; we show that such temporal modulation of selection pressure outperforms constant-dosage regimes.