Lower bound on the evolved fitness

The performance of a random library should give us a lower bound on the fitness of evolved libraries, given that I start the simulation with random antibody libraries. I therefore derived the expected fitness of a random library on the complete pathogen set of size 2^L. Let $\phi(p_i)$ be the score of an individual with respect to pathogen p_i and m the number of matching bit positions between a pathogen and an antibody. For a pathogen binding to a single random antibody, the probability that there are x or fewer matching bits, $Pr\{m \leq x\}$, is given by the value of the cumulative binomial at x. If we have A random antibodies, the probability that all of them have x or fewer matching bit positions with the pathogen is $\left[Pr\{m \leq x\}\right]^{A}$. Then the probability that the score $\phi(p_i)$ of the individual with respect to pathogen p_i is x/L, is given by the probability that at least one antibody has x matching sites with the pathogen but none has more than x, i.e.,

$$Pr\{\phi(p_i) = x\} = \left[Pr\{m \leq x\}\right]^{A} - \left[Pr\{m \leq x-1\}\right]^{A}.$$

The expected score of a random library of A antibodies with a random pathogen p_i is then given by

$$E[\phi(p_i)] = \frac{1}{L} \sum_{x = 0}^L x Pr\{\phi(p_i) = x\}.$$

The expected score of a random library on a randomly chosen pathogen p_i also represents the expected score of a random library over the complete set of 2^L pathogens. We then denote the expected fitness of a random library over the complete pathogen set by f_r,

$$f_r = E[\phi(p)] = \frac{1}{L} \sum_{x = 0}^L x \left[Pr\{m \leq x\}\right]^{A} - \left[Pr\{m \leq x-1\}\right]^{A}.$$ (2.1)

The above equation for f_r gives a lower bound on the fitness of the evolved libraries as a function of L and A.