Hamming distance mutation

Relatedness refers to the Hamming distance between master sequence and mutant and is expressed by the number of mutation events that are required to produce the mutant from the master. The frequency of individual mutants in the quasispecies is determined by their fitness and the Hamming distance from the master sequence. 

If genotypes are ordered in sequence space, the support forms an area which consists of one, two or more connected compounds. T vo genotypes are connected when they are separated by a single point mutation, i.e. when they have Hamming distance one. 

Several types of autocorrelation are often used for landscapes. In several important papers, Weinberger and Stadler consider both autocorrelation between adjacent points along a random walk in the landscape and autocorrelation between points a given Hamming distance apart independent of any walk [67,77,78,82,83], Both definitions yield similar information about the landscape and can be computed from one another for stationary landscapes. Other types of autocorrelation are based on neighborhoods defined by complex mutation operations such as crossover [45-49,85],... 

Fig. 4. The role of neutral networks in evolutionary optimization through adaptive walks and random drift. Adaptive walks allow to choose the next step arbitrarily from all directions where fitness is (locally) nondecreasing. Populations can bridge over narrow valleys with widths of a few point mutations. In the absence of selective neutrality (upper part) they are, however, unable to span larger Hamming distances and thus will approach only the next major fitness peak. Populations on rugged landscapes with extended neutral networks evolve along the network by a combination of adaptive walks and random drift at constant fitness (lower part). In this manner, populations bridge over large valleys and may eventually reach the global maximum ofthe fitness landscape.

Fig. 2.5. A quasi-species-type mutant distribution around a master sequence. The quasi-species is an ordered distribution of polynucleotide sequences (RNA or DNA) in sequence space. A fittest genotype or master sequence /m, which is commonly present at highest frequency, is surrounded in sequence space by a cloud of closely related sequences. Relatedness of sequences is expressed (in terms of error classes) by the number of mutations which are required to produce them as mutants of the master sequence. In case of point mutations the distance between sequences is the Hamming distance. In precise terms, the quasi-species is defined as the stable stationary solution of Eq. (2) [16,19, 20], In reality, such a stationary solution exists only if the error rate of replication lies below a maximal value called the error threshold. In this region, i.e. below...

An important feature of the replication-mutation kinetics of Eq. (2) is its straightforward accessibility to justifiable model assumptions. As an example we discuss the uniform error model [18,19] This refers to a molecule which is reproduced sequentially, i.e. digit by digit from one end of the (linear) polymer to the other. The basic assumption is that the accuracy of replication is independent of the particular site and the nature of the monomer at this position. Then, the frequency of mutation depends exclusively on the number of monomers that have to be exchanged in order to mutate from 4 to Ij, which are counted by the Hamming distance of the two strings, d(Ij,Ik) ... 

Within this model all mutation rates can be expressed in terms of only three quantities the chain length of the polymer, X, the single-digit accuracy of replication, q, often expressed as mutation rate per site and replication, p = 1 - q, and the Hamming distance, d(Ij,Ii). Finally, the (dependent) parameter, s = (1 - q)/q is the ratio between single digit mutation rate and accuracy. 

This conservation is a consequence of assumption (ii), namely, that mutants originate exclusively through erroneous replication and not through external interferences such as radiation or chemical attack. (If this assumption is relaxed, destruction terms must be subtracted in the conservation law to balance the additional first-order off-diagonal mutation terms.) The nondiagonal elements of the value matrix depend strongly on the Hamming distance d(i,k) between template i and erroneous replica k. For the uniform error rate model the expression reads... 

---