Sequence Hamming distance

Figure 9. Sequence spaces of biopolymers and hypercubes. Genotypes may be ordered according to their Hamming distances. The sequence spaces of binary...

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. 

Fig. 2. Geometric representation of a sequence space. Shown is the sequence space for a dinucleotide using a four-letter alphabet. Each point represents a sequence. Each double-arrowed line represents a Hamming distance h = 1.

Since this representation of sequence space would be extremely complicated to view and apprehend, we choose to use a simplified cartoon. The cartoon representation of a fitness landscape (Fig. 3) used in this chapter has only three dimensions. The z-axis scales the relative fitness of molecules. Fitness is arbitrarily defined, and may pertain to how well an RNA binds to a particular ligand, or how well it catalyzes a desired reaction. The sequences on the top of the cartoon peaks have a higher degree of fitness than the ones on the plateau. The xy-plane represents a two-dimensional apparition of much more complex sequence spaces. Sequences that have smaller Hamming distances would be closer... 

Fig. 11. The local fitness distributions around fourteen representative wild types. The curves were determined analytically for the fully additive landscape by Aita and Husimi for sequence length N = 60 and alphabet size A = 20. Each wild type is shown at the center of the concentric circles. The axes y is the scaled fitness (= F/ sN, s is the mean of F and here is negative) and x is the scaled Hamming distance from the optimum (= do/N). Each local fitness distribution is expressed as a concentric pie chart showing the fraction of mutants having Ay between l/N and (/ + 1)/N, where l — — 5, —4, — 3,. . . , 4. The thick curves represent the contours satisfying Ay = 0. Reprinted from Aita and Husimi (1998a) with permission, 1998 by Academic Press.

Hamming distance The Hamming distance between two sequences of equal length is the number of character positions in which they differ. The Hamming Distance is a distance measure. 

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...

The equation expresses that the space of all genotypes, the sequence space I, is a discrete space with the Hamming distance as metric. It is mapped onto a discrete space of structures called shape space with the structure distance as metric (We use I rather than 4 in order to indicate different numbering schemes used for sequences and structures). The evolutionarily relevant quantity, the fitness value fk as shown in Fig. 2.3, is derived from the phenotype Sk through evaluation, which can be understood as another mapping, a map from shape space into the positive real numbers including zero, fk = f(Sk). Both maps need not be invertible in the sense that more than one phenotype may have the same fitness value, and more than one sequence may lead to the same structure. We shall study here neutrality induced by the first map, (// in Eq. (6). 

Connectedness of a neutral network, implying that it consists of a single component, is important for evolutionary optimization. Populations usually cover a connected area in sequence space and they migrate (commonly) by the Hamming distance moved. Accordingly, if they are situated on a particular component of a neutral network, they can reach all sequences of this component. If the single component of the connected neutral network of a common structure spans all sequence space, a population on it can travel by random drift through whole sequence space. 

One approach to calculating the stationary mutant distributions for longer sequences is to form classes of sequences within the quasi-species. These classes are defined by means of the Hamming distance between the master sequence and the sequence under consideration. Class 0 contains the master sequence exclusively, class 1 the v different one-error mutants, class 2 all v(v —1)/2 two-error mutants, and so on. In general we have all (JJ) fe-error mutants in class k. In order to be able to reduce the 2 -dimensional eigenvalue problem to dimension v 1, we make the assumption that all formation rate constants are equal within a given class. We write Aq for the master sequence in class 0, Ai for all one-error mutants in class 1, 4 2 for all two-error mutants in class 2, and in general A for all k error mutants in class k. 

The two master sequences have a Hamming distance d(l, 2) = 2. The normalized concentrations are then... 

The two master sequences have Hamming distances d(l, 2)>3. This leads to selection in the limit considered ... 

Figure 13. Neighborhood relations in sequence space. In A, B, and C we show neighborhoods of pair of sequences (/[, 12) with Hamming distances d(l, 2) of 1, 2, or 3, respectively (v = 6). Part drawn in thick lines is general connections in thin lines depend on chain length v. To give example, for v = 5 one connection of equivalent set has to be eliminated. (D) Assignment of numbers to individual sequences of sequence space for v = 5 shown as used in Figures 14-30.

In the limit q->0 we observe an analogous situation. Here the degenerate master sequences are to be replaced by degenerate master pairs. A pair of complementary sequences always has a Hamming distance d(/, / -) = v. As the Hamming distance between two pairs we use the smallest of the three... 

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