Mutant fitness distribution

Mean-field theory can be used to predict the effects of mutation rate and parent fitness on the moments of the mutant fitness distribution (Voigt et al, 2000a). In this analysis, only the portion of the mutant distribution that is not dead (zero fitness) or parent (unmutated) is considered. The mutant effects are averaged over the transition probabilities without the cases of mutations to stop codons or when no mutations are made on a sequence. In order to obtain the fitness distribution, two probabilities are required (1) the probability pi(a) that a particular amino-acid identity a exists at a residue i, and (2) the transition probability that one amino acid will mutate into another Q = 1 — (1 — pm)3. The probability vectors p a) can be determined through a mean-held approach (Lee, 1994 Koehl and Delarue, 1996 Saven and Wolynes, 1997). The amino acid transition probabilities Q are calculated based on the special connectivity of the genetic code and the per-nucleotide mutation rate. Removing transitions to stop codons and unmutated sequences only requires the proper normalization of the probabilities pi and the moments. For example, the first moment of the fitness improvement w of the uncoupled fitness function is written as... 

Fig. 15. The (a) mean and (b) standard deviation of the mutant fitness distribution as the sequences ascends the fitness landscape. The data are shown for a smooth (A) and rugged (O) landscape.

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.

The population of mutant states in the quasi-species is strongly modulated by the fitness distribution. The effect is particularly strong for those mutants (i) whose fitness values resemble closely that of the wild type (m) due to the hyperbolic form WaH IV — fVn). Moreover, the population number of a given mutant is related not only to the wild-type fitness but also to the fitness values of its neighboring mutants, especially those situated in the subspace referring to all positions that differ from the master sequence. If, in such a domain, all fitness values (or a major portion thereof) are close to that of the master sequence, there will be a tremendous amplification of population numbers relative to domains of low fitness. Optimization routes are to this degree deterministically ordained. 

Mean-field theory can be used to predict the effects of mutation rate, landscape ruggedness, and parental fitness on the moments of the mutant fitness distribution1791. In this analysis, only the portion of the mutant distribution that is not dead (zero fitness) or parent (unmutated) is considered. The mutant effects are averaged over the transition probabilities. In order to obtain the fitness distribution, two sets of probabilities are required (1) the probabilities P (o) that a particular amino acid identity o exists at a residue i, and (2) the transition probabilities that one amino acid... 

Myelin in situ has a water content of about 40%. The dry mass of both CNS and PNS myelin is characterized by a high proportion of lipid (70-85%) and, consequently, a low proportion of protein (15-30%). By comparison, most biological membranes have a higher ratio of proteins to lipids. The currently accepted view of membrane structure is that of a lipid bilayer with integral membrane proteins embedded in the bilayer and other extrinsic proteins attached to one surface or the other by weaker linkages. Proteins and lipids are asymmetrically distributed in this bilayer, with only partial asymmetry of the lipids. The proposed molecular architecture of the layered membranes of compact myelin fits such a concept (Fig. 4-11). Models of compact myelin are based on data from electron microscopy, immunostaining, X-ray diffraction, surface probes studies, structural abnormalities in mutant mice, correlations between structure and composition in various species, and predictions of protein structure from sequencing information [4]. 

The conditions under which a population approaches a stationary, i.e. time independent, mutant distribution were derived from the kinetic differential equations. In this stationary distribution called quasispecies, the most frequent genotype of highest fitness, the master sequence, is surrounded by closely related mutants1 (Figure 10). 

The result is true for most fitness landscapes and seems to hold for all realistic landscapes in molecular evolution. There are, however, very smooth distributions of fitness values sometimes used in population genetics for which the transition between stationary quasispecies and drifting populations is smooth. A simple landscape showing a sharp transition is the single-peak fitness landscape that assigns a higher fitness value to the master sequence and the same lower fitness value to all mutants. It has some similarity to mean field approximations often applied in physics. 

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. 7. The probability distribution P(c) of a positive mutation with c coupled interactions occurs as the sequence ascends the fitness landscape (generated using a spin-glasslike fitness function). The distribution is shown at two positions on the fitness landcape, a random sequence (O) and a highly optimized sequence (A). As the sequence is optimized, the probability that positive mutations will be made at uncoupled residues increases considerably. The mutation rate is an average of one amino acid (three nucleotides) per sequence and the number of mutants screened is 3000. Reprinted from Voigt et at. (2000b), with permission.

Figure 11.14. Distributions of (logarithmsof)the apparent open time (left)and apparent shut time (right) for wild-type human receptors (top) and for mutant eL221F receptors (bottom). The histogram shows the experimental observations. The continuous lines were not fitted directly to the data in the histograms, but were calculated from the rate constants for the mechanism that was fitted (Fig. 11.10, scheme B, with the two sites constrained to be independent). The distributions were calculated with appropriate allowance for missed events (HJC distributions) (65, 66). The fact that they superimpose well on the histograms shows that the mechanism was a good description of the observations. The dashed lines show the distributions calculated from the fitted rate constants in the conventional way (45), without allowance for missed events, so they are our estimate of the true distributions cf open and shut times (296). See color insert.

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