Analytical model for vaccination protocols that optimally produce broadly neutralizing antibodies

One way that the adaptive immune system responds to infectious pathogens is by creating antibodies (Ab) that can bind specifically to the associated antigens (Ag). In order to generate such Abs, B cells go through many rounds of Darwinian mutation and selection during the affinity maturation (AM) process. Successful vaccination guides the AM to produce B cells that elicit neutralizing Abs against the pathogen of concern. For highly mutable pathogens such as HIV, however, B cells that respond to the Ags presented during natural infection or vaccination generally neutralize a small number of mutant strains. The desired outcome of vaccination in these cases is to generate optimal amounts of the so-called broadly neutralizing Abs (bnAbs) that protect against various strains of the fast-mutating pathogen. Our goal is to describe the mechanisms via which bnAbs might be elicited by properly designed vaccination procedures. We devise a minimal model of the B cell population dynamics that focuses on their mutations and also the selection forces imposed by the vaccine. Using an analytical approach based on operator formalism, we show that to maximize the bnAb production, the selection forces imposed by sequential vaccination rounds over time need to become more focused on the B cells that have a chance to reach the high breadth state by mutation only. We also investigate how the initial distribution of the B cells may modify the optimal vaccination protocol.