Sequence distribution statistics

The Markovian character of the sequence distribution statistics in the macromolecules results [6, 94] from assumption about the steady-state of the radical concentrations, which usually holds with a high degree of accuracy in the copolymerization processes [6, 95], It is worth mentioning that along with such kinetic stationarity one should usually speak about the statistical stationarity. It means that when the number of the units in copolymer molecules exceeds 10-15, their composition practically becomes independent on degree of polymerization and is indistinguishable from the value predicted by the stationary Markov chain theory. This conclusion is supported by the theoretical [96,97,6] and experimental [98] evidence. 

MMA polymerization is one of the most studied systems and was thought to be explicable, within experimental error, in terms of Bemoullian statistics. Moad et ai.jb have made precise measurements of the configurational sequence distribution for PMMA prepared from 13C-labeled monomer. It is clear that... 

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

Eigenvalues of the operator Qr are real while the largest of them, Af, equals unity by definition. As a result, in the limit n-> oo all items in the sum (Eq. 38), excluding the first one, Q Q f = Xr/Xfh will vanish. In this case, chemical correlators will decay exponentially along the chain on the scale n 1/ In AAt values n < n the law of the decay of these correlators differs, however, from the exponential one even for binary copolymers. This obviously testifies to non-Markovian statistics of the sequence distribution in molecules (see expression Eq. 11). The closer is to unity, the greater are the values of n. The situation when n 1 corresponds to proteinlike copolymers. 

Now let us examine the distribution and position of disulfides in proteins. The simplest consideration is distribution in the sequence (see Fig. 51), which is apparently quite random, except that there must be at least two residues in between connected half-cystines. Even rather conspicuous patterns such as two consecutive halfcystines in separate disulfides turn out, when the distribution is plotted for the solved structures (Fig. 51), to occur at only about the random expected frequency. The sequence distribution of halfcystines is influenced by the statistics of close contacts in the three-dimensional structures, but apparently there are no strong preferences of the cystines that could influence the three-dimensional structure. 

A strategy that attempts to obtain a statistically unbiased sample or data set for a series of experimental measurements by simulating a chance distribution or chance sequence. See Statistics (A Primer)... 

Note In a random copolymer, the sequence distribution of monomeric units follows Bemoullian statistics. 

Statistical copolymers are copolymers in which the sequential distribution of the monomeric units obeys known statistical laws e.g. the monomeric-unit sequence distribution may follow Markovian statistics of zeroth (Bemoullian), first, second or a higher order. Kinetically, the elementary processes leading to the formation of a statistical sequence of monomeric units do not necessarily proceed with equal a priori probability. These processes can lead to various types of sequence distribution comprising those in whieh the arrangement of monomeric units tends towards alternation, tends towards... 

Statistical analysis of the stereochemical sequence distributions (Table 8-3 and Sec. 8-16) also supports the enantiomorphic site control model. 

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 ... 

From the analysis of 13C NMR spectra of polypropylenes, Doi96) found that the sequence distribution of inverted propylene units follows first-order Markov statistics. Table 4 lists the two reactivity ratios rQ and rt, for the polymerization of propylene with the soluble catalysts composed of VC14 and alkylaluminums at — 78 °C ... 

Initially, the protein-like HP sequences were generated in [18] for the lattice chains of N = 512 monomeric units (statistical segments), using for simulations a Monte Carlo method and the lattice bond-fluctuation model [34], When the chain is a random (quasirandom) heteropolymer, an average over many different sequence distributions must be carried out explicitly to produce the final properties. Therefore, the sequence design scheme was repeated many times, and the results were averaged over different initial configurations. 

The data shown in Tables HI and TV show that the 13C nmr spectra of vinyl chloride-vinylidene chloride copolymers have a redundancy of structural relationships. By analyzing a range of compositions, this system has been found to yield a reasonable description of both monomer composition and monomer sequence distribution. The data also show that this copolymer is a good example of a system best described by first order Markovian statistics as compared to Bernoullian statistics. 

Carbon-13 nuclear magnetic resonance was used to determine the molecular structure of four copolymers of vinyl chloride and vinylidene chloride. The spectra were used to determine both monomer composition and sequence distribution. Good agreement was found between the chlorine analysis determined from wet analysis and the chlorine analysis determined by the C nmr method. The number average sequence length for vinylidene chloride measured from the spectra fit first order Markovian statistics rather than Bernoullian. The chemical shifts in these copolymers as well as their changes in areas as a function of monomer composition enable these copolymers to serve as model... 

Such dependencies are quite usual for the terminal and penultimate models since in these cases the sequence distributions are described by Markov statistics [51-53, 6]. In the former case this description is carried out by means of the Markov chain, the states S of which correspond to the individual monomer units... 

When the Markov character of unit sequence distribution in the copolymer is established and the elements of matrix Q are known, the standard procedure of Markov chain theory allows one to obtain the explicit formulae for all the statistical characteristics of the copolymer fraction obtained at given monomer feed composition Xj by means of the simple algebraic operations [51-53, 6J. 

In contrast to the above mentioned models, the similar statistical description of the products of the complex-radical copolymerization occurring through the scheme (2.5) has been carried out quite recently [37, 49, 55-60]. Within the framework of this Seiner-Litt model, both copolymer composition [37,49, 55-58] and fractions of the different triads and blocks of the monomer units in the macromolecules were calculated [57]. The probability approaches which were applied in these works, are regarded as being of limited applicability in contrast to the general statistical method [49, 59, 60], By means of the latter method, the sequence distribution and composition inhomogeneity of the copolymer were completely described [49, 60] and also thorough calculations of its microstructure with the account for the tactidty were carried out [59, 60]. 

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