Conformational jump models

Dejean et al. [7], proposed another conformation jump model (referred to as the DLM model), in which a librational motion of the jump axis was introduced as the third motion. This model will correspond to the 3t model described below but the derivation of G (t) was empirically made in this case. 

Among the various expressions that are based on a conformational jump model and have been proposed for the orientation autocorrelation function of a polymer chain, G t), the formula derived by Hall and Helfand (HH) [4] leads to a very good agreement with fluorescence anisotropy decay data. It is written as... 

Our data of T in the coil form in Figure 10 can nearly be put over the master curve for T of polystyrene in the organic solvents, as shown in Figure 11, with the vfflues of the scaling parameters p and q p = 1.305 and q = 0.045(K ), by which the data of T y P in the coil form (in Figure 10) can also be put over the master curve for T P of polystyrene in the organic solvents. Therefore, the molecular motion of (MA-St)p in the coil form may be described in terms of the conformationed jump model combined with isotropic rotational diffusion, when the ratio Xp/Xp seems to be nearly 0.07. 

Fig. 20. (a) Allowed side chain conformations, (b) Distribution of CK-2H bond vectors, (c) Three-site jump model as an approximation of multi-site jumps. Reproduced with permission from the Society of Polymer Science, Japan. 

For main-chain acrylic radicals, created in solution at room temperature and above, the presence of a superposition of conformations or Gaussian distributions is unlikely. Polymers undergo conformational jumps on the submicrosecond timescale, even in bulk at room temperamre. ° The first two theories above require that the radicals be fairly rigid with little (Gaussian distribution) or no (superposition of static conformations) movement around the Cp bond. The main-chain radical is sterically hindered but still quite flexible, and a dramatic change in the hybridization at Ca is unlikely. We have approached our simulations with the hyperfine modulation model. 

The dynamics of bulk polymers have been approached in two different ways. On one hand, models of localized conformational jumps have been proposed to interpret numerous NMR experiments (see e.g. Ref. or . These models, which are specific of a given polymer assume that a short chain sequence performs conforpia-tional jumps between a few number of sites, the rest of the chain being immobile. Such localized jumps would lead to a well separated elastic peak in neutron quasielastic scattering experiments, in contradiction with all the experimental data obtained from polymer meltsIndeed, these models can in some cases be invoked to describe secondary relaxations in glassy polymers, but they are not sufficient to account for the numerous liquid-like properties of polymer melts. 

By comparison of observed and theoretically calculated spectra it can be shown that these carbons are involved in gauche-trans conformational jumps of the C-D bond through a dihedral angle of 103°, and from the correlation times as a function of temperature an activation energy of 5.8 kcal/mol is found. Several seemingly plausible motional models are excluded by these results, but the data agree with models proposed by Helfand (21,22) for motion about three bonds. 

The detailed analysis of carbon-13 spin-lattice relaxation times of a number of polymers either in solution or in bulk at temperatures well above the glass-transition temperature has led to a general picture involving several types of motions. The segmental reorientation can be interpreted in terms of correlated conformational jumps which induce a damped orientation diffusion along the chain. It is satisfactorily described by the well-known autocorrelation functions derived from models of conformational jumps in polymer chains [4,5] which have proven to be very powerful in representing fluor-... 

Weber-Helfand model the primary parameter is the correlation time for cooperative backbone transitions, X]. At the lower temperatures studied, xq plays an increasing role in the Weber-Helfand model but xj is still the major factor. This is an interesting point in itself since cooperative transitions were also found to predominate when the Weber-Helfand model was applied to the polycarbonates. Here in the polyformal, single bond conformational transitions do play a larger role and this can be seen in the three bond jump model as well by the drop of m to 1 at lower temperatures. Since xj and x are both measures of the time scale for cooperative motions, it is interesting to note that the Arrhenius summaries of the two correlation times in Tables II and III are very similar. This similarity, taken together with the domination of cooperative transitions in the Interpretations, supports the utility of both models though the Weber-Helfand model is developed from a more detailed analysis of chain motion. 

The assumption of free rotation about each C-C bond in an alkyl chain can give conformations of molecules that are precluded on grounds of excluded-volume effects. Following Tsutsumi, a jump model was employed [8.11] to describe trans-gauche isomerisms in the chain of liquid crystals by allowing jumps about one bond at any one time. To evaluate internal correlation functions gi t), not only the equilibrium probabilities of occupation given by Eq. (8.4) are needed, but also the conditional probability P(7, t 7o,0), where 7 and 70 denote one of the three equilibrium states (1, 2, 3) at times t and zero, respectively,... 

Let us now use the jump model to calculate the average in equation (8) when the vector from i to j undeigoes jumps among N states. The probability that the system is in the jUth conformer is assumed to follow the rate (or master ) equations ... 

Crystallographers have had some success in using normal mode descriptions of internal motion, with effective frequencies as adjustable parameters, and initial steps have been taken to incorporate similar ideas into NMR refinements. This model builds in important large-scale correlated motions, but not generally the effects of local conformational Jumps. 

Hart et a/. (1981) and P. A. Hart and C. F. Anderson (unpublished) have treated this problem using an approach analogous to that of Tsutsumi (1979). The very simplest model for internal motion that allows specification of conformational probabilities is the two-state jump model in which independent internal motion is manifested as a transition between two states populated unequally. While this model has not been applied to small oligonucleotides, it has found use in the rationalization of DNA restriction fragment relaxation times. The details of that work are discussed following the completion of this section. The two-state model is included here because it leads easily and naturally to the general form of Eq. (2). 

Fig. 11a and b. Decay of the alignment echo height as a function of the mixing time x2 for different motional mechanisms, a Tetrahedral jumps as a model for conformational changes b Diffusive motion, the solid lines correspond to unrestricted rotational diffusion, the dashed lines to diffusion restricted to an angular region of 8°. Note the strong dependence of the decay curves on the evolution time t, in case of diffusive motion... 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

---