Two-state jump model

Fig. 1. Line shape for the two-state-jump model. The frequencies are in units of being the frequencies of the two states. The numbers on curves indicate the modulation rate, a = y/toj.

As another simplification, we assume that the modulation 2 takes only two values co,. This is a generalization of the two-state jump model mentioned in Section 111. The basic space for Eq. (63) is then 2x2 dimension. It is convenient to write Eq. (63) as... 

Recently, the stochastic models for the Mossbauer line shape problem have been discussed by several investigators.20 Such models can be treated in a systematic way as we have described in the above. For example, in a 57Fe nucleus, the spin in the excited state is / = and that in the ground state is / = i, so that the Hamiltonian is a 6 x 6 matrix. If a two-state-jump model is adopted, the dimension of the matrix equation, Eq. (63), is 6 x 2 = 12. If the stochastic operator is of the type (26), then the equation is a set of six differential equations. These equations can be solved, if necessary, by computers to yield the line shape functions for various values of parameters. 

Fig. 15.—Bistable (two-state) jump model. [Reproduced with permission from Fig. 3 of P. Dais, Carbohydr. Res., 160 (1987) 73-93 and Elsevier Science B.V.]...

The parameter t is given in Eq. A-9 in the Appendix, as a function of the correlation time, t associated with internal motion. One of the input parameters is the angle j3, formed between the relaxation vector (C—H bond) and the internal axis of rotation (or jump axis), namely the C-5—C-6 bond. The others are correlation times t0 and r, of the HWH model, obtained from the fit of the data for the backbone carbons. The fitting parameters for the two-state jump model are lifetimes ta and tb, and for the restricted-diffusion model, the correlation time t- for internal rotation. The allowed range of motion (or the jump range) is defined by 2x for both models (Eqs. A-4 and A-9). 

Two-State Jump Model (Isotropic Overall Motion)58... 

Q and the Three-Time Correlation Function Two-State Jump Model Exact Solution... 

Figure 4.1. Spectral trails of a SM undergoing a very slow spectral diffusion process described by the two-state jump model are shown for (a) W (without the shot noise) and (b) n (with the shot noise). Parameters are chosen as G = 0.2F, R = 10 F, v = 5F, F =...

Note that these two approximations, Eqs. (4.72) and (4.73), are not limited to the two-state jump model but are generally valid in the fast modulation regime. The frequency correlation function is given by... 

Table IV.2 summarizes various expressions for the line shape and Q found in the limiting cases of the two-state jump model investigated in this work. For simplicity, we have set fj = 1. We see that although the fluctuation model itself is a simple one, rich behaviors are found. We believe that these behaviors are generic (although we do not have a mathematical proof). In the weak modulation cases, both and Q can be described by a single expression, irrespective of the fluctuation rate R. However, in the strong modulation cases, and Q change their qualitative features as R changes.

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). 

Two cases can be distinguished for jump models. In the first case, the dipolar-coupled atoms are bonded to each other. Further, the axis around which jumps occur must intersect or be rigidly attached to one of the atoms. This case is a propos of relaxed by directly bonded protons and has been treated by London (1978). Seveitd jump models have appeared (e.g., Woessner, 1962a Marshall et al., 1972 Hubbard and Johnson, 1975). In this case, as in the models for free and restricted difiusion, the coordinate system can be defined such that only angle yo is time-dependent. In efiect, the drawing in Fig. 4a is also applicable for a two-state jump model but the two states exist only with angles +yo and —yo, not intermediate values. The spectra] density for dipolar relaxation in tl case is (London, 1978, 1980)... 

---