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A persistent public health challenge is identifying immunization schemes that are effective against highly mutable pathogens such as HIV and influenza viruses. To address this, we analyze a simplified model of affinity maturation, the Darwinian evolutionary process B cells undergo during immunization. The vaccination protocol determines the selection forces that steer affinity maturation to generate antibodies. We focus on identifying the optimal selection forces exerted by a generic time-dependent vaccination protocol to maximize the production of broadly neutralizing antibodies (bnAbs) that can protect against a broad spectrum of pathogen strains. The model utilizes a path integral representation and operator approximations within a mean-field limit and provides guiding principles for optimizing time-dependent vaccine-induced selection forces to enhance bnAb generation. We compare our analytical mean-field results with the outcomes of stochastic simulations, and we discuss their similarities and differences. 

As a population grows, spreading to new environments may favor specialization. In this paper, we introduce and explore a model for specialization at the front of a colony expanding synchronously into new territory. We show through numerical simulations that, by gaining fitness through accumulating mutations, progeny of the initial seed population can differentiate into distinct specialists. With competition and selection limited to the growth front, the emerging specialists first segregate into sectors, which then expand to dominate the entire population. We quantify the scaling of the fixation time with the size of the population and observe different behaviors corresponding to distinct universality classes: unbounded and bounded gains in fitness lead to superdiffusive (z = 3/2) and diffusive (z = 2) stochastic wanderings of the sector boundaries, respectively. 

In growing populations, the fate of mutations depends on their competitive ability against the ancestor and their ability to colonize new territory. Here we present a theory that integrates both aspects of mutant fitness by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (Kardar-Parisi-Zhang equation). We solve these equations and find three regimes, which are controlled solely by the expansion rates, solely by the competitive abilities, or by both. Collectively, our results provide a simple framework to study spatial competition.