Lower bound on the fitness

Let us first determine the fitness of a random library as a function of the library size. I will write the derivation in the most general sense, in terms of the density distribution of the bond strength, g(x), and its corresponding cumulative density function, G(x), and I will apply it to the particular Gaussian distribution of bond strengths that I mentioned above.

For every pathogen, the fitness is given by the maximum of A random variables drawn from the distribution G, A being the size of the antibody library. The probability that the bond strength between a random pathogen and all of the antibodies in the library is less than or equal to a value, x, is G(x)^A, and then the derivative of this, giving the probability density of fitness x, is

$$g_A(x) = \frac{d}{dx} \left[(G(x))^A \right] = A \times g(x) (G(x))^{A-1}.$$ (2.3)

Now the expected fitness of a random library of A antibodies on the complete pathogen space, given the probability density function of the fitness, g_A(x), is

$$f_g(A) = \int_0^\infty x g_A(x) dx = \int_0^\infty x \frac{d}{dx}\left[G_A(x)\right] dx.$$ (2.4)

Let y = G_A(x), taking values between 0 and 1. Then $\frac{d}{dx} \left[G_A(x)\right] = dy$ and Eq. can be rewritten in terms of y as

$$f_g(A) = \int_0^1 x(y) dy,$$ (2.5)

where x(y) denotes the fact that x has to be expressed now as a function of y. But y = G_A(x) = (G(x))^A, thus $G(x) = y^{\frac{1}{A}}$, and $x = G^{-1}(y^{\frac{1}{A}})$, where G^-1 denotes the inverse function of G. With this, Equation becomes

$$f_g(A) = \int_0^1 G^{-1}(y^{\frac{1}{A}}) dy.$$ (2.6)

In the case of the Gaussian-distributed bond strengths, mentioned above, we cannot derive an analytical form for the fitness dependency on antibody library size, as it is impossible to analytically invert the normal distribution. We may, however, compute the values numerically, and this is the approach that I used in generating the data for random antibody libraries shown in Fig. . As mentioned above, for the case that I studied, the bond strengths are Gaussian distributed, with mean 0, and variance 20.