Conformational statistics

The prime objective of this concise review is to provide an illustration of the interaction of these two disciplines using particular examples. In choosing the examples, we seek to demonstrate the potentialities of the conformation-dependent design of the sequences of monomeric units in heteropolymer macromolecules. Under such a design, their chemical structure is controlled not only by the kinetic parameters of a reaction system but also by the conformational statistics of polymer chains. 

The first of these assumptions, generally accepted in macromolecular chemistry [1,3], is correct enough when considering the propagation reaction under copolymerization of the majority of monomers. Simple estimates reported in paper [74] support the correctness of the second assumption. As for the third one, it is true, strictly speaking, only under 0-conditions. The conformational statistics of macromolecules in a thermodynamically good solvent is known [30] to differ from the Gaussian one. Nevertheless, this distinction may hardly influence the qualitative conclusions of the simplest theory of interphase copolymerization. To which extent the account of the excluded volume of macromolecules will affect quantitative results of this theory, may be revealed exclusively by computer simulations. 

In the same year in which Stoll et al. (1934) defined the cyclisation constant C, Kuhn (1934) laid the foundations of the theoretical approaches to the conformational statistics of hydrocarbon chains and considered the cyclisation probability of the chain as a fundamental, chain length dependent phenomenon related to chain shape. He proposed to view the specific rate kintra of an intramolecular reaction between a pair of reactive groups attached to the ends of a chain molecule as the product of the effective concentration Ceff of... 

Fattal, D. R. and Ben-Shaul, A. (1994). Mean-field calculations of chain packing and conformational statistics in lipid bilayers comparison with experiments and molecular dynamics studies, Biophys. J., 67, 983-995. 

It is however possible to find conditions, called unperturbed or theta conditions (because for each polymer-solvent pair they correspond to a well-defined temperature called d temperature) in which a tends to 1 and the mean-square distance reduces to Q. In 6 conditions well-separated chain segments experience neither attraction nor repulsion. In other words, there are no long-range interactions and the conformational statistics of the macromolecule may be derived from the energy of interaction between neighboring monomer units. For a high molecular weight chain in unperturbed conditions there is a simple relationship between the mean-square end-to-end distance < > and the mean-... 

More elaborate models, from the point of view of conformational statistics, lead to analogous conclusions (373-375). Even admitting the possibility of in-... 

The conformational statistics of asymmetric vinyl chains such as P2VN are well-known 126). The rotational conformers of isotactic (meso) dyads are entirely different from those of syndiotactic (dl) dyads. Frank and Harrah132) have described each of the six distinct conformers for meso and dl dyads, using the t, g+ and g nomenclature of Flory 126). Excimer-forming sites (EFS) are found in the tt and g g+ meso states, and in the degenerate tg , g t dl state. Because the rotational conformers of compounds such as l,3-bis(2-naphthyl)propane do not match those of either the iso-or syndiotactic dyads of P2VN, the propane compounds make poor models of aryl vinyl polymers. However, the rate constants of fluorescence and decay of the intramolecular excimer in polymers can usually be determined from the propane compounds (but see the exceptional case of PVK and its models133)). 

The three-state RIS model of conformer statistics is used to analyze the 16 independent dipole coupling constants measured in a proton NMR study of n-hexane in a nematic liquid crystal solvent. The orientational ordering of the n-hexane molecule is treated in the context of the modular formulation of the potential of mean torque. This formulation gives an accurate description of alkane solute orientational order and conformer probabilities in the nematic solvent. Consequently, substantially more accurate calculated diplar couplings are obtained, and this is achieved without the need to resort to unconventionally high values of the trans-gauche energy difference E(g) in the RIS model. 

The probability distribution of isomeric conformations in PDMS is investigated by both conformational energy considerations and by molecular dynamics simulations. A comparatively smooth distribution of isomeric states is obtained from both approaches. A new RIS treatment, compatible with the molecular mechanics and dynamics considerations, is introduced for describing the conformational statistics of PDMS. 

N 051 "Conformational Statistics of Short Chains of Poly(L-alanfne) and Potygfycine Generated by Monte Carlo Method and the Partition Function of... 

N 144 "Conformational Energy Calculations on Alginlc Acid II. Conformational Statistics of the Copolymers"... 

As a consequence of the single chain approximation the conformation statistics of any two chains are independent of each other. Correspondingly, the mean autocorrelation function is given by... 

Next, we examine the term i2. In a gas-like single segment approximation, this term can be replaced by 1212. The molecular conformation statistics are independent of each other. This might be due to the fact that in the absence of a three-dimensional lattice-potential, nematic shifts of neighboring segments are very likely to occur. In this approximation the configuration does not depend on which individual pair of molecules k, 1 is picked out The molecular structure factor is independent of the indexes k and L Hence 1 inter, d can be written as... 

Chen, S. J. (2008). RNA Folding Conformational statistics, folding kinetics, and ion electrostatics. Annu. Rev. Biophys. 37, 197—214. 

Deviations of (r02) ( ) M can also be brought about by other causes, for example by chain branching as an extreme case we can think of a highly complex branched structure in which the many branches are again branched, so that it can hardly be defined as a chain. The conformation statistics is then no longer applicable, and for such polymers a value of a near to 0 has even be found this means that V ( ) M, just as with small molecules. Viscosimetric determination of M is then no longer applicable, since [q] is independent of M. 

Application of conformation statistics results in a value of ASm. Combination with AHm gives the Flory-Huggins relation ... 

Using the generator-matrix formalism of the chain conformational statistics, the chain conformation partition function E can be expressed as a product of three matrices... 

The methods of conformational statistics, discussed so far, had as starting point the real polymer chain. The aim was to relate the dimensions of the coiled polymer molecule statistically to the mutual displaceability of the chain atoms. Nearly exact relationships are obtained for a large number of freely jointed or freely rotating elements. Under conditions of restricted movability, however, the statistical equations can generally not be solved and empirical factors like s, a and a are introduced. 

For polyelectrolytes, the way In which the conformation statistics are computed from the field is the same as discussed in secs. 5.5a-f. The new feature originates from the electrostatic Interactions, which affect the field. We make a distinction between strong polyelectrolyte segments which have a fixed (pH-independent) charge and weak groups that adjust their charge to the (local) pH. For the former case the situation is relatively simple we may extend the field as given in (5.5.5) with an electrostatic term 17 ... 

M. Vacatello and D. Y. Yoon, Macromolecules, 25, 2502 (1992). Conformational Statistics of Polyisobutylene by a Monte Carlo Study. 

D.A. Brant, A.E. Tonelli and P.J. Flory, The conformational statistics of random poly(lactic acid) chains. Macromolecules, 2 (1969) 228... 

Finally, we note that Schmidt-Rohr and co-workers have elegantly demonstrated that 2D 13C-13C DQ spectra recorded for static samples can identify the chain conformation statistics for 13C-labeled polymer samples.112 Remembering that the frequency of a given 13C resonance depends on the orientation of the CSA tensor, the method relies on the fact that the adoption of a particular torsional angle along the chain results in DQ peaks for only specific pairs of 13C frequencies. In particular, it was shown that trans and gauche conformations lead to very different 2D DQ powder spectra, and it was thus possible to quantitatively determine the conformation statistics for a sample of amorphous polyethylene terephtha-late) (PET). 

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