Conformation correlation function

In contrast with these models, which start from a rather crude representation of the chain, but allow a direct computation of the OACF, Hall and Helfand proposed recently a model able to predict conformational correlation functions (CCF) for rather realistic molecular potentials. The OACF cannot be derived from these CCF, at least at the present time. However, Hall and Helfand suggested that the CCF for a chain of two-state elements ... 

Conformational transition rates can be obtained from computer simulations by simple counting, hazard analysis, or monitoring the decay of conformational correlation functions [39, 40]. The definition of a transition is somewhat arbitrary and different investigators have made different choices. One could... 

Comparison with Results of Brownian Dynamics (BD) Simulations 2.2.2.1 Conformational Correlation Functions... 

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

The method is likely to be useful for the numerical calculation of other correlation functions of importance to complex molecules. An example is the orientation correlation functions of interest in NMR-derived dynamical estimates for proteins and nucleic acids [134], Such correlations are difficult to converge numerically when multiple conformations separated by large free energy barriers contribute to their measurement. 

The definition of correlation functions in this book differs from the definition of the correlation coefficient in the theory of probability. The difference is essentially in the normalization, i.e., whereas g(, ) can be any positive number 0 S g the correlation coefficient varies within [-1,1]. We have chosen the definition of correlation as in Eq. (1.5.19) or (1.5.20) to conform with the definition used in the theory of liquids and solutions. 

It is seen that the correlation function g(l, 1) is not simply related to the direct correlations and Clearly, this is not an average of the two direct correlations [see also Eq. (4.5.24) below]. In this section we wish to focus on the indirect correlation, Therefore, for the moment, we assume that the direct correlations are either negligible, i.e., 5 5 1, or that they are independent of the conformation, i.e.,S, = 5 , = 5. Hence, g(l, 1) may be written as... 

Perhaps the simplest two-site cooperative systems are small molecules having two binding sites for protons, such as dicarboxylic acids and diamines. Despite their molecular simplicity, most of these molecules do not conform with the modelistic assumptions made in this chapter. Therefore, their theoretical treatment is much more intricate. The main reasons for this are (1) there is, in general, a continuous range of macrostates (2) the direct and indirect correlations are both strong and intertwined, so that factorization of the correlation function is impossible. In addition, as with any real biochemical system, the solvent can have a major effect on the binding properties of these molecules. 

An example that conforms to this definition would be benzene-l,3,5-tricar-boxylic acid (Section 5.9). Clearly, owing to the symmetry of the molecule there is only one intrinsic binding constant k, and only one intrinsic binding constant for pairs fcj, or, equivalently, only one pair correlation function... 

The Hb molecule consists of four subunits two a-subunits (each with 141 amino acid residues) and two P-subunits (each with 146 amino acid residues). The distances between the subunits in the two conformations of the Hb are shown schematically in Fig. 6.1b. The tetramer as a whole has a roughly spherical shape. Since the four subunits are not identical, one cannot expect that the four binding sites will be strictly identical. Nevertheless, in most of the theoretical treatments of the binding properties of Hb, one assumes that the sites are nearly identical, hence all of the intrinsic binding constants, as well as the correlation functions, must be understood only in an average sense, as discussed in Appendix J. 

Accounting for molecular conformations or torsional and rotational chain dynamics, it is more useful to calculate the mean intramolecular structure factor in terms of density correlation functions (DCF). The structure factor results simply from a Fourier transform of the corresponding DCF. 

Another way of obtaining the characteristic time scale and dynamical range of conformational dynamics is from the equilibrium correlation functions of the FRET efficiency ... 

This correlation function is shown under different conditions in Fig. 5. In the folded state, the decay of the correlation function is characterized by a relatively narrow dynamical range and short time scales, on the order of 1. The rapid decay of C(t) in this case can be attributed to fluctuations around the native conformation, within the corresponding basin of attraction. The fact that those fluctuations are fast is consistent with the fact that this is a rapid two-state folder. More specifically, the native conformation is expected... 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

---