Hamming distance distributions

Fig. 18. Distribution of fitnesses as a function of mutant Hamming distance from current position on the landscape for an initial fitness of (a) 0.5 and (b) 0.543. Simulations were carried out on NK landscapes with N= 100 and K = 2, yielding the high nearest neighbor correlation of 0.97 and a correlation length of 33.3. Vertical bars show +1 and -1 standard deviation from the mean of fitnesses found at each search distance. If the best of six mutants at each distance is chosen, then the best mutant can be found at Hamming distance 33 from the fitness 0.5 point and at decreasing distances as the initial fitness increases. (From Ref. 119.)...

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.

Aita and Husimi (1996) proposed that the additive model can be applied to give a rough estimate for the Hamming distance from the wild type to the optimum, the fitness slope near the wild type, and the height of the optimum. They calculated the expected fitness distribution and compared this to experimental data produced by the mutagenesis of a region of E. coli lac promoter. Based on a fit between the theoretical and experimental distributions, they estimated that the Hamming distance between the wild type lac promoter and the optimum is seven to ten amino acid substitutions and the activity could be improved 100 to 1000-fold. 

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...

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. 

Weight distribution The list of the Hamming distances of each codeword from a given reference codeword. 

If A requires k bits and we encode values with a maximum size of n bits, we need n + k bits to store encoded values. Assuming a failure model with equally distributed bit flips and that the Hamming distance between all code words is constant the resulting probability p of not detecting an error is p = "TumbefoTpotTbi rdt = 2- = Thus, the error detection capability is... 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

Azulene has weak absorption in the visible region (near 7000 A) and more intense band systems in the ultraviolet. The first ultraviolet system, which commences at about 3500 A, has been examined in substitutional solid solution in naphthalene (Sidman and McClure, 1956) and in the vapour state (Hunt and Ross, 1962), and can be observed in fluorescence from the vapour (Hunt and Ross, 1956). Theory predicts that the transition is 1Al<-lAl(C2K), i.e. allowed by the electronic selection rules with polarization parallel to the twofold symmetry axis (see, e.g., Ham, 1960 Mofifitt, 1954 Pariser, 1956b). The vibrational analysis shows that the transition is allowed but does not establish the axis of polarization. The intensity distribution among the vibrational bands indicates a small increase in CC bond distance without change in symmetry. 

---