Error Threshold (evolution) - A Simple Mathematical Model Illustrating The Error Threshold

A Simple Mathematical Model Illustrating The Error Threshold

Consider a 3-digit molecule where A, B, and C can take on the values 0 and 1. There are eight such sequences (, and ). Let's say that the molecule is the most fit; upon each replication it produces an average of copies, where . This molecule is called the "master sequence". The other seven sequences are less fit; they each produce only 1 copy per replication. The replication of each of the three digits is done with a mutation rate of μ. In other words, at every replication of a digit of a sequence, there is a probability that it will be erroneous; 0 will be replaced by 1 or vice versa. Let's ignore double mutations, and divide the eight molecules into three classes depending on their Hamming distance from the master sequence:

Hamming distance	Sequence(s)
0
1
2
3

Note that the number of sequences for distance d is just the binomial coefficient for L=3, and that each sequence can be visualized as the vertex of an L=3 dimensional cube, with each edge of the cube specifying a mutation path in which the change Hamming distance is either zero or ±1. It can be seen that, for example, one third of the mutations of the molecules will produce molecules, while the other two thirds will produce the class 2 molecules and . We can now write the expression for the child populations of class i in terms of the parent populations .

where the fitness matrix w is given by:

where is the probability that an entire molecule will be replicated successfully. The eigenvectors of the w matrix will yield the equilibrium population numbers for each class. For example, if the mutation rate μ is zero, we will have Q=1, and the equilibrium concentrations will be . The master sequence, being the fittest will be the only one to survive. If we have a replication fidelity of Q=0.95, then the equilibrium concentrations will be roughly . It can be seen that the master sequence is not as dominant. If we have a replication fidelity of Q=0, then the equilibrium concentrations will be roughly . This is almost an equal population of all sequences. (If we had perfectly equal population of all sequences, we would have populations of /8.)

If we now go to the case where the number of base pairs is large, say L=100, we obtain behavior that resembles a phase transition. The plot below on the left shows a series of equilibrium concentrations divided by the binomial coefficient . (This multiplication will show the population for an individual sequence at that distance, and will yield a flat line for an equal distribution.) The selective advantage of the master sequence is set at a=1.05. The horizontal axis is the Hamming distance d . The various curves are for various total mutation rates . It is seen that for low values of the total mutation rate, the population consists of a quasispecies gathered in the neighborhood of the master sequence. Above a total mutation rate of about 1-Q=0.05, the distribution quickly spreads out to populate all sequences equally. The plot below on the right shows the fractional population of the master sequence as a function of the total mutation rate. Again it is seen that below a critical mutation rate of about 1-Q=0.05, the master sequence contains most of the population, while above this rate, it contains only about of the total population.

It can be seen that there is a sharp transition at a value of 1-Q just a bit larger than 0.05. For mutation rates above this value, the population of the master sequence drops to practically zero. Above this value, it dominates.

In the limit as L approaches infinity, the system does in fact have a phase transition at a critical value of Q: . One could think of the overall mutation rate (1-Q) as a sort of "temperature", which "melts" the fidelity of the molecular sequences above the critical "temperature" of . For faithful replication to occur, the information must be "frozen" into the genome.