Statistical weight sequence

For both cases, the assumption is valid that only one helical sequence exists and that products with the same number of hydrogen bonds have the same stability. Considering the statistical weights of the possible intermediates, the whole measurable degree of conversion, t0,ai, is computed by the mass-action law and can be derived from Eq. (6)149. ... 

In a first approximation, interdependence among the rotations around successive bonds is expressed for polyethylene by putting the statistical weight of the G G sequences equal to zero. With respect to the preceding model a further increase in the value of is then observed. A more detailed analysis (157)... 

The parameter s is the statistical weight of a helix unit inside a helical sequence, or it may be interpreted as the equilibrium constant for the growing reaction of a helical sequence. The free energy associated with it thus may be called the transition free energy for a given polypeptide-solvent pair. When, as usual, it is decomposed as... 

The parameter statistical weight of a terminal unit of a helical sequence. The AG associated with it is the free energy required to create a helix unit located at the boundary between a helical sequence and a random-coil one. Usually, a is called the cooperativity parameter for the formation of helical sequences. The reason will be explained in Section B.3. 

The shortest helical sequence that can be created in an a-helix-forming polypeptide is hhh, and its statistical weight is as. Since s is of the order of unity in the helix-coil transition region, the probability that such a nucleus for the growth of a helical sequence will be produced is essentially equal to a. For this reason, a is also called the helix-initiation parameter. 

It is investigated whether the stereochemical 13C NMR chemical shifts in the resonance peaks can be ascribed to differences in the conformations in the various stereoisomers. The authors follow Boyd and Breitling (A 022) in thair statistical treatment of the PP chain, with the exception that here conformational sequences are not excluded of the type XG/G Y for two adjacent diads unless XG or G Y imply another c (syn-axiall interaction within either diad. Therefore this treatment, which is more rigorous but consistent with Boyd and Breitling s energy calculations, requires statistical weights which are functions of three adjacent torsional angles. 

Mean-square unperturbed dimensions a and their temperature coefficient, d tin 0) I d T, are calculated for ethylene-propylene copolymers by means of the RIS theory. Conformational energies required in the analysis are shown to be readily obtained from previous analyses of PE and PP, without additional approximations. Results thus calculated are reported as a function of chemical composition, chemical sequence distribution, and stereochemical composition of the PP sequences. Calculations of 0 / nP- are earned out using ( ) r r2 = 0.01, 1.0, 10.0, and 100.0, (ii) p, = 0.95, 0.50, and 0.05, liii) bond length of 153 pm and bond angles of 112°for all skeletal bonds, iv) = 0 and 10°, and (v) statistical weight factors appropriate for temperatures of 248, 298, and 348 K. Matrices used are ... 

For molecules limited to one helical sequence, the basic statistical weight matrix used is given by ... 

The statistical weight for amino acid residue / is unity if it is not in a helical state. Its statistical weight is os if it initiates a sequence of helical amino residues and s if it propagates an existing helix. 

With these introductory remarks we can state the approach of current efforts to compute the conformations of polypeptides of known amino acid sequence. First of all, it is necessary to obtain an expression for the energy of the system (protein plus solvent) as a function of the coordinates of the atoms of the system. Secondly, with the aid of a computer, the energy must be minimized. Finally, the nature of the energy surface in the neighborhood of the minimum must be explored to obtain the statistical weight. 

Figure 10. Quasi-species as function of single-digit accuracy of replication (q) for chain v = 5. We plot relative stationary concentration of master sequence ( (,),fum of relative stationary concentrations of alt one-error mutants ((i), of all two-error mutants ( j), etc. Note that we have only one five-error mutant 7,5, = /s, in this particular example. We observe selection of master sequence at g = 1. Then relative concentration of master sequence decreases with decreasing q. At value q = 0.5 all sequences are present in equal concentrations. Hence, sums of concentrations of two- and three-error mutants are largest—they have statistical weight of 10—those of the one-and four-error mutants are half as large—they have statistical weight of 5—and finally master sequence 7q and its complementary sequence, the five-error mutant /ji, are present in relative concentration ofonly. At q = 0 we have selection o( master pair", which consists of/o and /31 in our example. Thus we have direct replication with errors in range 1 > g > 0.5 and complementary replication with errors in range 0 < q < 0.5. Rate constants chosen as Aq = 10[U ] and A = 1 [t ] for all mutants Ic 0. Here we denote arbitrary reciprocal time unit by [t" ]. All degradation rate constants were put equal 7>o = D, = Dj = = D31 = 0.

Let the polymer chain be capable of assuming two different conformational states, coil and helix, the latter capable of binding iodine as I or I. Thus an amylose molecule may be described as a sequence of coil and helix sites, which for present purposes can each be taken to comprise approximately six glucose residues. Coil states can be denoted by c, unbound helix states by h, and helix states bound either to I or I by b. We define "complex" as any uninterrupted sequence of b states, regardless of length. Then in the usual fashion (3 ) we define the statistical weight matrix U indexed for sites i-1 and i as... 

Two matrices that describe conformations that alternate in sequence are used to specify conformation in tactic vinyl polymers. Tacticity introduces asymmetric elements into the chain. One convention used in specifying chain asymmetry consists of distinguishing between right handed chains termed d and left handed or 1 chains. This convention, admittedly arbitrary, is borrowed from the field of optical activity. For a tactic polymer with a threefold potential (Figure 2.13), the statistical weight matrix for a d placement between neighboring bonds is... 

The partition function is obtained by first forming the product of statistical weights U for each chain conformation and then summing over all possible conformations. As written, equation 2F.1 would tax the capacity of high-speed computers for even short chains N < 20 atoms). However, by using the RIS approximation, the evaluation of equation 2.F.1 becomes tractable. Essentially, the sequence of operations described by equation 2.F.1 is reversed (i.e., the conformational statistical weights are first suimned and then the sums are multiplied). Present day computers can readily handle this revised sequence. 

When pseudoasymmetric centers are used for the description of the stereochemical sequence, six distinct statistical weight matrices, Eqs. (3.18)-(3.20), are required. They can be replaced by a total of three statistical weight matrices, denoted by Up,Um, and Ur, if the stereochemical sequence is described instead as a sequence of meso and racemo diads. 

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