  Coevolutionary arms races form between interacting populations that
constitute each other's environment and respond to mutual changes. This
inherently far-from-equilibrium process finds striking manifestations in the
adaptive immune system, where highly variable antigens and a finite repertoire
of immune receptors coevolve on comparable timescales. This unique challenge to
the immune system motivates general questions: How do ecological and
evolutionary processes interplay to shape diversity? What determine the
endurance and fate of coevolution? Here, we take the perspective of responsive
environments and develop a phenotypic model of coevolution between receptors
and antigens that both exhibit cross-reactivity (one-to-many responses). The
theory predicts that the extent of asymmetry in cross-reactivity is a key
determinant of repertoire composition: small asymmetry supports persistent
large diversity, whereas strong asymmetry yields long-lived transients of
quasispecies in both populations. The latter represents a new type of Turing
mechanism. More surprisingly, patterning in the trait space feeds back on
population dynamics: spatial resonance between the Turing modes breaks the
dynamic balance, leading to antigen extinction or unrestrained growth. Model
predictions can be tested via combined genomic and phenotypic measurements. Our
work identifies cross-reactivity as an important regulator of diversity and
coevolutionary outcome, and reveals the remarkable effect of ecological
feedback in pattern-forming systems, which drives evolution toward non-steady
states different than the Red Queen persistent cycles.
