Fig. 2.

The optimal cost and receptor distributions for protecting against a one-dimensional Gaussian antigenic landscape $Q\left(a\right)$ of variance ${\sigma }_Q^2$, as a function of the cross-reactivity width σ. As σ increases, the optimal distribution $P\ast \left(r\right)$ becomes narrower and narrower (Left and Center Insets), until it concentrates entirely onto a single point, for $\sigma \ge \sqrt{2}{\sigma }_Q$ (Right Inset). The minimal cost (multiplied by σ for a comparison at constant recognition capability) is constant below the transition point, but increases with σ past it. The cross-reactivity function, which quantifies the affinity between receptor r and antigen a as a function of their distance in shape space, has a Gaussian form, $f\left(r-a\right)=\text{exp}\left[-{\left(r-a\right)}^2/2{\sigma }^2\right]$, and the cost function is linear in the effective recognition time, $F\left(m\right)=m$.