Partition function, conformational

In the RIS model for PE described by Abe et al. [143], the conformational partition function, Z, of the unperturbed chain with n bonds is formulated using a well-known statistical weight matrix for bond i, Uj, that is usually written in a 3 x 3 form. 

RIS approach consists of a factorization of the conformational partition function into local contributions. The conformational partition function of a simple linear chain consisting of skeletal bonds 1 < i < n is given in most cases by... 

A matrix formulation of the conformational partition function is used to assess the influence of irregular structures on the formation of intramolecular antiparallel (5-sheets. An antiparallel sheet is considered to be irregular If any pair of contiguous strands has an unequal number of residues. The regular structures in the model consist of antiparailel sheets in which every strand contains the same number of residues. The regular structures in the model consist of antiparailel sheets in which every strand contains the same number of residues. Aside from a growth parameter r, the model contains two parameters, 8 and t, that account for the influence of edge effects. 

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 effect of the interdigitation between the polymer chains of two brushes on the interaction force profile is plotted in Fig. 8 for three polymers PVP-PI 69-39, PVP-PI 38-69, and PS-X (140 K). The solid curves present the results in which the interdigitation was taken into account, while the dashed ones those in which the interdigitation was neglected. For the former, the chain conformational partition function [Eq. (5)] involves a 2mX2m supermatrix G for which the chain free ends can be located in any lattice layer between the two surfaces. For the latter, the conformational partition function involves a m X in supermatrix G with chains con-... 

Prompers and Briischweiler showed by quasiharmonic analysis that the conformational partition function of a globular protein sampled on the ns time scale can be factorized in good approximation into purely reorientational part, which determines heteronuclear NMR spin relaxation, and a remaining part that includes other types of intramolecular motions. Thus a thermodynamic interpretation of NMR relaxation parameters in proteins in the presence of motional correlations can be given. 

The necessity for a proper averaging over a multitude of conformations remains when the protein is partially ordered. The power of the matrix methods can still be utilized if the dominant ordering interactions are of relatively short range, as they are in helix formation (6). The conformational partition function, Z, for a chain of n residues is formulated as shown in Equation 1. 

In Section II we show how characteristic packets and macrostate annealing can be used to efficiently compute thermodynamic integrals over complicated potentials, and in Section III we show how the same ideas can be used to efficiently compute intermacrostate transition rates. In Section IV we show how this approach can be adapted for analyzing macromolecular integrals and demonstrate it on the pentapeptide Met-enkephalin. We summarize and discuss the next steps to be taken in Section V. For simplicity we consider the evaluation of the conformational partition function Zc, but the method also applies to averages such as Eqs. (1.5). Although we focus on proteins, it should be possible to apply a similar approach to problems... 

Flory (1969) has stressed that the conformational partition function for a polymer molecule is extremely large (e.g. of order 2 even for a short chain of 100 bonds). For this reason, a polymer chain has many ways of fulfilling its conformational proclivities in the presence of a second chain. Consequently, interpenetration is strongly favoured over denting. 

Ch. 6) and its combinatorial part conformational partition function for the C, N, I phases... 

The conformational partition function for tactic vinyl polymers using two statistical weight matrices is given by... 

The rotational state parameters are adjusted until the best agreement with several types of experimental data is obtained. The RIS approximation permits the use of matrix methods based on the following considerations. The conformational partition function (Qc) (Appendix 2A) is required for averaging over all possible conformations of a polymer chain. 

The energetics of a polymer chain conformation can be readily specified by its partition function (see Appendix 2A). In the rotational state approximation, the conformational partition function ( gc) can be expressed as the product of the statistical weights for each bond (JJ) fi-om 2 to n- summed over all rotational states. Thus... 

Z = the conformational partition function Pi - statistical weight matrices A = the value of the largest eigenvalue... 

Zq the eornponent conformational partition function per chain regardless of... 

Further, h, is obtained from equation [5.2.17], but Zg is the conformational partition function for a copolymer with at Tg and Z is the component conformational partition function for a copolymer with Xa regardless of the temperature in Z. Whereas for binary random copolymers, Tg is given by ... 

As we shall see below, the denominator in Eq. (3.5) is the conformational partition function, Z , for the RIS model of the chain. It is constructed as a sum of Boltzmann factors that depend on T and the energies of the first- and higher-order interactions present in all of the conformations of the chain. Structural information does not appear explicitly in Z . However, a wealth of structural information (/, 6, (f>) can appear in the numerator of Eq. (3.5). The numerator also contains all of the thermal and energetic information from Z . The combination of this information allows a rapid estimation of C , even at large n, because computers can rapidly calculate the serial matrix products that appear in the numerator and denominator of Eq. (3.5). 

The conformational partition function, in the RIS approximation, is the sum of the statistical weights for the conformations in the RIS model. The terminal row and column vectors must be formulated so that they will extract the desired sum of the statistical weights of all conformations firom V2D2. .. V iD 1. 

The conformational partition function is subject to the same types of manipulations as are other partition functions encountered in statistical mechanics. Thus the average conformational energy of the chain is obtained from the temperature dependence of Z . 

The previous portion of this chapter has focused primarily on the use of the RIS model for the computatimi of the mean square unperturbed dimensions, because that is the most frequent application of the model. This section describes briefly many other applications of the RIS model. AU of these applications employ the conformational partition function, but the additional information incorporated in the calculation, and the manipulation of Z, depend on the appUcatimi. 

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