The fitness of evolved libraries

How does the fitness of the evolved libraries compare to the bounds that we calculated?

I used a string length L = 8 bits to explore the scaling relation between the maximum fitness evolved by a library and the number of antibodies, A, in the library. The fitness of the libraries was computed over the complete set of 2^L pathogens at each time step. For each library size, I averaged the best fitness values obtained in 10 independent runs.

 

Fig. shows the fitness of the best evolved library $\bar{f}$, as well as the upper (f_u) and lower (f_r) bounds that I calculated. It is clear that the fitness, $\bar{f}$ increases slower than logarithmically as a function of the library size. In fact, we may infer the following approximate relation:

$$f = c \log^\alpha(A).$$

$\bar{f}$, f_u and f_r obey roughly the same scaling relation, with $\alpha \approx 0.2$. The c values however differ between the different curves, and the basis for this difference is explored below. The evolved libraries do not always reach the fitness f_u as given by the "perfect" libraries, but they generally come very close to this value. The similar scaling relation for evolved libraries, as well as for random libraries and "perfect" libraries suggests that the scaling relation is mostly a result of the geometry of the bit-string space and the additive nature of the matching rule.

Thus to obtain an increase of $\delta f$ in fitness one would have to multiply the size of the libraries by larger and larger factors. The selection pressure for increasing the size of the germline-encoded repertoire is thus expected to be progressively lower. A similar dependency was suggested, on experimental grounds, and within a somewhat different model, by Minar (1994).