Quasispecies Model - Mathematical Description

Mathematical Description

A simple mathematical model for a quasispecies is as follows: let there be possible sequences and let there be organisms with sequence i. Let's say that each of these organisms asexually gives rise to offspring. Some are duplicates of their parent, having sequence i, but some are mutant and have some other sequence. Let the mutation rate correspond to the probability that a j type parent will produce an i type organism. Then the expected number of i type organisms produced by any j type parent is ,

where .

Then the total number of i-type organisms after the first round of reproduction, given as, is

Sometimes a death rate term is included so that:

where is equal to 1 when i=j and is zero otherwise. Note that the n-th generation can be found by just taking the n-th power of W substituting it in place of W in the above formula.

This is just a system of linear equations. The usual way to solve such a system is to first diagonalize the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is irreducible, then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium.