. 2015 Apr 27;112(19):5950–5955. doi: 10.1073/pnas.1421827112

Table 1.

Cost functions and optimal repertoires

$F (m)$	$P_{r}^{*}$	$A ({\tilde{N}}_{a})$
$m^{α}$	$C Q_{r}^{1 / (1 + α)}$	$C' {(N_{st} / {\tilde{N}}_{a})}^{1 + α}$
$\ln m$	$C Q_{r}$	$C' (N_{st} / {\tilde{N}}_{a})$
$1 - exp (- β m)$	$max {C \sqrt{Q_{r}} - β, 0}$	$C' / {(β + {\tilde{N}}_{a} / N_{st})}^{2}$
$Θ (m - m_{0})$	$max {ln (Q_{r}) / m_{0} - C, 0}$	$C' \exp (- m_{0} {\tilde{N}}_{a} / N_{st})$

The cost function F(m) measures the harm caused to an organism by the time that immune receptors have had m encounters with a pathogen. The optimal receptor distribution $P *$ is determined by minimizing this cost, given a pathogen distribution Q and a cross-reactivity function $f_{r, a}$ specifying the probability that receptor r binds to antigen a. The second column gives the form of $P *$ over scales larger than the cross-reactivity. The optimal $P *$ can be reached as a steady state resulting from competitive binding between receptors and antigens (last section of Results) quantified by an “availability function” A. ${\tilde{N}}_{a} = \sum_{r} N_{r} f_{r, a}$ represents the coverage of antigen a by the repertoire; $N_{st} = \sum_{r} N_{r}$ is the total steady-state population; and $C, C', β, and m_{0}$ are positive constants.