Sequence space fitness value

Fig. 2.3. Sequence space and genotype - phenotype mappings. Mapping genotypes onto phenotypes and into fitness values. The sketch shows a map from sequence (or genotype) space onto phenotype space, as described in the text, and further into the real numbers resulting in fitness values assigned in two steps to the indi-...

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). 

Figure 21. Mutant space high-value contour near local optimum. Diagram is multiply branched tree with different macromolecular sequences at vertices. Each line joins neighboring sequences whose values are within 0.5 of locally optimum sequence at lower center for linearized fitness function of type 2 [Eqn. (IV.7)] and reference fold that is cruciform, like tRNA, for sequence of length 72. Over 1300 branches shown extending up to 10 mutant shells away from central optimum. Better sequence (labeled optimum) was found in tenth mutant shell. Non-random sampling of mutant sequences demonstrated typical of population sampling in quasispecies model. Note small number of ridges that penetrate deeply into surrounding mutant space. (Additional connected paths due to hypercube topology of mutant space not shown.)...

If sufficient numbers of diffraction peaks are observed, simple inspection for the absence or weakening of particular peaks in an observed sequence of d spacings can provide a useful estimate of the radius (or thickness) of domains. To obtain accurate values, however, it is necessary to compare the whole of the observed intensity profile (after correcting for polarization, the Lorentz factor, etc.) with intensity profiles predicted by theoretical models. When there are only a few peaks (which is almost always the case with spherical domains, and is often the case with cylindrical domains) then a fitting procedure must be used. With the scattering profile obtained for the polystyrene-6/ock-polybutadiene-Wock-polystyrene (10000 71000 10 000 M ) copolymer this approach yielded a value of 12 nm for the cylindrical diameter. 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2,..., 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly... 

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