Shape space coverage with other matching rules

Although the concept of a shape space has spun numerous studies on the behavior and evolution of the immune system, it is not clear that intermolecular interactions are well described in this manner. In fact, a survey of the literature also reveals discussions of the relevance of the shape-space model, at least in idiotypic interactions (1994). I therefore decided to investigate the impact of another fitness function on the basic scaling result that I obtained above. The fitness function that stems from the shape-space metaphor is highly structured, the fitness of an individual being given by the antibody with the smallest Hamming distance from the pathogen. We would like to know what happens if the fitness landscape has a completely different structure. The option I explore is based on the idea of a random energy model, introduced by Derrida (1984),in the context of spin-glasses.

If we view the antigen-antibody interaction from a biochemical standpoint, the strength of the bond is given by the difference of the free energies of the complex, and of the two molecules in their unbound state. A realistic representation of the energy landscape as a function of the sequence of the molecules is clearly impossible at this point. Therefore I use the following abstraction. I assume that each molecule has an "energy", which is a random deviate from a Gaussian distribution. The antigen-antibody complex also has an energy corresponding to it, which is a random deviate of a Gaussian distribution. The difference between the energy of the complex and the energy of unbound molecules gives the strength of the bond between them. I perform this calculation for all antibodies that the individual can make, and I take the maximum bond strength between an antibody and the pathogen to be the fitness with respect to that pathogen. I then use the evolutionary algorithm that I described in section to evolve libraries of different sizes on a complete pathogen set of size P = 2⁹ = 512. As the bit-strings that I used have length L = 9, the 7 high order bits are set to 0. The best library evolved in 1000 steps is used to infer the scaling relation between fitness and antibody library size.

The energy of antigens and antibodies is drawn from a Gaussian distribution with mean 50, and variance 2.5, whereas the energy of the complex was chosen from a Gaussian distribution with mean 100 and variance 10. Although the exact choice of the mean and variance of the energy of an individual molecule is arbitrary, there clearly is a scaling of the energy of a molecule with its size, so we expect that by doubling the size of the molecule we roughly double the energy associated with it. To determine the energy of each molecule, I seed the random number generator with the numerical representation of the bit string representing that molecule, and then calculate a pseudo-random Gaussian deviate according to the algorithm given in Press et al. (1988). I assign such an energy to both antigen and antibody. To obtain the antigen-antibody complex, I take the XOR between the bit strings representing the antigen and the antibody. I use the numerical representation of this bit string to calculate an energy, as described above. The bond strength, given by the difference in energy between the complex and the unbound molecules, will be distributed as a Gaussian with mean 0 and variance 15.

One might argue that the landscape thus constructed does not have any obvious structure for the evolutionary algorithm to work with, given that the energies assigned to closely related genotypes are random deviates from the Gaussian distribution. The landscape does, however, have some structure, as the antibodies with high energy have a better chance of lowering this energy by binding to pathogens. These are, in fact, the antibodies that the evolutionary algorithm discovers.

In the previous section I showed that, for the shape space model, the scaling relation between fitness and library size in the case of evolved libraries is essentially a shifted variant of the relation that we obtain for a random library of identical size. I will show that this is also the case for the energy model that I just described.