Fig. 5.

The immune repertoire can self-organize to a state that minimizes cost and provides protection against infections via competitive evolution of receptor populations stimulated by antigens. Numerical simulations of the population dynamics, as well as its mean-field limit (Eq. 2), show how competition causes a random initial receptor distribution to fragment into a highly peaked pattern [Insets represent $P_r\left(t\right)=N_r\left(t\right)/{\displaystyle {\sum }_r\prime N_r\prime \left(t\right)}$]. Top Right Inset represents the antigenic environment $Q_a$ driving the dynamics [generated from a lognormal noise of power spectrum $\propto 1/\left(1+{\left(5q\right)}^2\right)$ and coefficient of variation 1]. Departure from optimality, as measured by the relative cost gap $\left[\langle F\rangle \left(P_r\left(t\right)\right)-\langle F\rangle \left(P_r^{\mathrm{*}}\right)\right]/\langle F\rangle \left(P_r^{\mathrm{*}}\right)$, decreases with time and eventually reaches zero in the mean-field limit. The three independent runs of the stochastic dynamics show reproducible results. We use the availability function $A\left(\tilde{N}\right)=1/{\left(1+\tilde{N}/N_0\right)}^2$ with $N_0={10}^6$, a death rate $d=0.001$, and a cost function $F\left(m\right)=1-e^{-\beta m}$ with $\beta =1/110$. The space size is $10\sigma$. The initial condition was drawn from a lognormal noise of power spectrum $\propto 1/\left(1+{\left(5q\right)}^2\right)$, with coefficient of variation 2 and ${\displaystyle {\sum }_r{N}_r\left(0\right)}=1.1\times {10}^8$. In the stochastic simulations, the time between antigen presentations is $\mathrm{\Delta }t=0.005d^{-1}$ (200 infections per cell lifetime).