Model

To address these questions, I implemented an evolutionary algorithm, similar to the one introduced by Hightower (1996). The basic components of the model are the following:

• A population of individuals, each having a gene library of A genes. Each gene is represented by a bit string of length L. From this library, I assume that A antibodies are made, that is, all genes are expressed, and that all these antibodies are available for binding any of the pathogens. I do not distinguish between the genotype (antibody gene) and the phenotype (antibody molecule). One could, alternatively, view the libraries as representing the possible set of antibodies that an organism can produce. The genetic operators, to be discussed below, such as mutation and recombination on these libraries would then have to be thought to represent phenotypic changes to the antibody repertoire as a result of implicit genetic operations on the level of the genes. Also, I will not include the rearrangement process in the model. This choice is meant mainly to keep the model simple. Note, immune receptor rearrangement does not play a major role in generating diversity in all species, and thus the simple model that I propose has direct significance for these situations.
• Pathogens are also represented as bit strings of length L.
• The essence of the complicated antibody-pathogen interaction in the real world, that I want to capture in this model, is that for each pathogen in the environment, the host can raise at least one antibody that can recognize the pathogen. The level of recognition may or may not be protective for the individual. As suggested recently (1998), I will not require a certain affinity threshold for protection. I will simply assume that the lower the affinity, the lower the survival chances when the host is presented with that given pathogen. Thus, to each individual library, $\mathcal{A}$, I assign a score in matching a pathogen p, defined as
  
  $$\phi(p) = \frac{1}{L} (L - \min_{a \in \mathcal{A}} \left[ h(a,p) \right])$$
  
  

  where h(a,p) is the Hamming distance between antibody a and pathogen p. In other words, for each pathogen, we find the antibody with the minimal Hamming distance to the pathogen. The score is a number between 0 and 1, being maximal for a perfect match, at Hamming distance 0, and minimal for the case of complementary bit strings. Note that I use identical lengths for the antibody and the pathogen strings and that the bit strings are aligned prior to calculating the Hamming distance.

• In Hightower (1996), the fitness f of an individual was defined as the average score $\langle \phi \rangle$ with respect to all pathogens that it encountered. I will use the same fitness function here. I believe that this choice can be most generally justified in terms of the survival probabilities of an individual with respect to the pathogen challenges it encounters. All these challenges have to be met successfully if the organism is to survive. Let us assume that the probability s_p of surviving the attack of pathogen p grows exponentially with the score $\phi(p)$. That is, for each additional matching bit between the best antibody and the pathogen, the probability s_p that the organism survives goes up by a constant factor, $k^{\frac{1}{L}}$. Thus,
  
  $$s_p \propto k^{\phi(p)}.$$
  
  
  The probability of surviving all pathogen attacks is given by the product of the survival probabilities s_p for all pathogens p. Therefore, the total survival probability s is given by

  
  

  $$s \propto k^{P \langle \phi \rangle}$$
  
  

  where P is the number of pathogens, and $\langle \phi \rangle$ is the score of the library averaged over all pathogens. Thus, we find that the survival probability s is a monotonically increasing function of the average score $\langle \phi \rangle$. For the selection scheme (described below) that I used, only the relative ranking of the fitnesses of different libraries is important. Therefore, under the assumption that the fitness of an individual depends only on its survival probability s, we can identify the fitness with the average score $\langle \phi \rangle$. Formally, if we denote the pathogen set by $\mathcal{P}$, the fitness f of an individual is given by
  

  $$f = \frac{1}{P} \sum_{p \in \mathcal{P}} \phi(p) \equiv \langle \phi \rangle.$$
  
  

• I will evolve the antibody libraries on the following pathogen sets:
  • The complete set of 2^L pathogens of length L.
  • Random subsets $\mathcal{P}$ of the complete pathogen set of size 2^L. These sets are constructed by sampling P pathogens, with replacement, from the complete pathogen set.
  • Pathogen sets that evolve independently of the hosts.
• The evolutionary algorithm that I used has the following structure.   The initial population consists of M = 50 random libraries, of identical size, A. This population size is sufficiently large to allow convergence to relatively high fitness solutions, given the mutation rate of 0.002 per bit that I used in evolving the libraries. Each individual, then, consists of a single library. I use rank selection as follows: If r is the rank of the fitness of an individual in the population, the chance of that individual being selected as a parent is, on average, $w_r = \frac{2 (M-r)}{M(M-1)}$. To create one library of the new generation, I select, with replacement, two libraries of the old population, then generate two new libraries by crossing over the two chosen libraries. The number of crossover points n is chosen from a binomial distribution with mean 0.01 A. This crossover scheme is more realistic for our purpose of modeling the evolution of gene libraries than other schemes that are described in the evolutionary algorithms literature (1996). The crossover points are chosen at the boundary between antibodies, so individual antibodies are not disrupted by crossover. I then choose one of the new crossover products, mutate it, and add it to the new population. 1000 generations of the genetic algorithm constitute a run. At the end of the run, I take the library with the highest fitness in the population, and use it to infer the scaling relation, as well as for analyzing the properties of antibodies that were evolved.

A note about the random number streams. The basic function of the random number stream returns a random deviate from a uniform distribution on the interval [0,1). The algorithm is given in Knuth (1973), and the implementation that I used was written by Terry Jones. This function can be used to generate random deviates of the uniform density function over any interval between 0 and any positive value.