A TWO-STATE MODEL

The Schrodinger equation Hij/ = Eij/ usually solved by expanding the unknown wave function in a series of complete basis set of states 4 i, where N 

For the sake of simplieity, in further analysis the functions will be assumed normalized and real. Then, for AT = 2 we have Hu = = 

A non-trivial solution of these secular equations exists only if the secular determinant satisfies 

After expanding the determinant, we obtain a quadratic equation for the unknown energy E  

After inserting the above energies into the secular equations we obtain the following two sets of solutions ci and C2 . 

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The term state rather than condition is often used in this context. However, the latter seems preferable in an introductory account. This is because the del Castillo-Katz mechanism is often described as a two-state model of receptor action, meaning here that the occupied receptor exists in two distinct (albeit interconvertible) forms, AR and AR, whereas three conditions of the receptor (R, AR, and AR ) have to be identified when applying the law of mass action to the binding of the ligand, A. 

The fraction of Pn conformer in denatured proteins can be roughly estimated from the work of Park et al. (1997). They analyzed the data of Drake et al. (1988) on poly(Lys) over the temperature range of —100 to +80°C plus their own data on the peptide AcYEAAAKEAPAKEAAAKANMHs at temperatures from 0° to 90°C. They used a two-state model, justified by the tight isodichroic points observed in each system, and derived limiting 222 nm ellipticity values of +9500 deg cm2 dmol-1 for the Pn conformation and —5560 deg cm2 dmol-1 for the high-temperature ensemble of conformers. This leads to Eq. (1) (Bienkiewicz etal., 2000) ... 

Figure 4.2 Protein transformations in reversed-phase chromatography for a two-state model. The native folded state can exist in either the mobile phase (Fm) or the stationary phase (Fs), as can the unfolded state (Um, Us). The equilibrium constants (k) for interconversions of the four species are indicated. (Reproduced from X.M. Lu, K. Benedek, and B.L. Karger, J. Chromatogr., 359 19 [1986]. With permission from Elsevier Science.)...

Most of the work published to date on molecular dynamic studies of interfacial electron transfer involves the simplified assumption of a two-state model for the electronic degrees of freedom. Consider an ion of charge qj near a solution/metal interface. As a result of electron transfer between the ion and the metal surface, the charge of the ion changes to qj. We will consider both forward and backward electron transfer and assume that = <7 - = -1, so that the forward reaction corresponds to a single electron transfer from the metal to the ion, for example + e ... 

Figure 4.27. A two-state model for succinic acid Cis with ( > = 0 and Trans with (]) = jt.

Figure 4.28. The skeleton model for succinic acid, P-alanine, and ethane diamine. The model is essentially the same as that described in Fig. 4.27. Instead of a two-state model, we allow a continuous range of variation, 0 < > 2it. Also, and can be either negative or zero for a carboxylate or an amine group, respectively.

Figure 5. Free energy diagrams showing the salient differences between the Monod and Koshland models. The MWC model is a two-state model with equivalent ligand binding interactions (indicated here by the equal spacing between Rq and Rb states and between RLi and RL2 states). In the KNF model, the amount of energy released determines whether binding will be independent or show negative or positive cooperativity.

The self-assembly of this fundamental building block of chromatin is a topic of enduring interest. DNase I digestion experiments as well as spectroscopic studies indicate that nucleosome core particles can be reconstituted by salt-jump (i.e., diluting NaCl concentration from 2.0 to 0.2 M) or by direct mixing of histones and DNA at the lower salt concentration. Daban and Cantor used the increase in eximer fluorescence to investigate the reassembly process in terms of a two-state model ... 

Figure 8.7. Variation of AG with temperature for a two-state model (from Miodownik 1977).

For many nonpolar liquids, the electron drift mobility is less than 10 cm /Vs, too low to be accounted for in terms of a scattering mechanism. In these liquids, electrons are trapped as discussed in Sec. 4. Considerable evidence now supports the idea of a two-state model in which equilibrium exists between the trapped and quasi-free states ... 

Figure 11. (a) Bias as a function of the number of pulls N for a two-states model. The inset... 

The GMH method of Cave and Newton [39, 40] is based on the assumption that the transition dipole moment between the diabatic donor and acceptor states vanishes, i.e., the off-diagonal element of the corresponding dipole moment matrix is zero. Thus, in the localization transformation one diagonalizes the dipole moment matrix of the adiabatic states ij/i and ij/z. For a two-state model, the rotation angle ft) can be expressed with the help of the transition dipole moment and the difference of the dipole mo-... 

Since a comparison of various van t Hoff enthalpies (calculated by analysis of da/dT at Tm, the slope of /Tm vs In cjot. and the shape of the calorimetric curve)0 with the calorimetric enthalpy shows that A// h A//cai, it was concluded that the melting transition in this duplex is cooperative and well represented by a two-state model. 

While we have identified some of the design principles for ft in a two-state model, design criteria for molecules with enhanced 7 appear more difficult to state succinctly because single large matrix elements do not necessarily collapse the 7 expression to... 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

A two-state model of solubilization may be used to describe the location of solutes in micellar systems. This model involves a distribution between a dissolved state, which is associated with the core, and an adsorbed state, associated with the micellar water interface. The molecules in thi dissolved state remain in the micelle because of the solvent properties of the core. Molecules in the adsorbed state are due to the surface activity of the dissolved species, similar to a surface exces (Mukerjee, 1979). 

S. Wilson, K. Jankowski, and J. Paldus, Int.]. Quantum Chem., 28,525 (1985). Applicability of Non-Degenerate Many-Body Perturbation Theory to Quasi-Degenerate Electronic States. II. A Two-State Model. 

We now briefly summarize the key results of the analysis of Refs. [50,51] for a reduced XT-CT model of the TFB F8BT heterojunction, using explicit quantum dynamical (MCTDH) calculations for a two-state model parametrized for 20-30 phonon modes. At this level of analysis, an ultrafast ( 200 fs) XT state decay is predicted, followed by coherent oscillations, see Fig. 8 (trace exact in panel (a)). Further analysis in terms of an effective-mode model and the associated HEP decomposition (see Sec. 4.2) highlights several aspects ... 

These early studies also showed that the sharp change in the CH band frequency at the cmc can be explained by a two-state model. The linear combination of two synthetic Lorenztian bands, separated in frequency by much less than their... 

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