The two-state model

For a two-state transition between a native state, N, and an unfolded state, U, the equilibrium unfolding constant, Kud, partition coefficient, Q, mole fraction of each state, X, and standard free energy change for the unfolding reaction are defined as follows. 

The value of JCun (or AG will depend on the extent of progress along the perturbation axis (temperature, T, pressure, P, pH, etc.). The following sections give various functions that are generally accepted as describing the dependence of AG°un on these perturbation axes. One of the equations below, when combined with those above and equation 1 can describe spectral data as a function of temperature or chemical denaturant, in terms of the two-state model. 

This model has an exact solution (even for a large perturbation V). One may introduce various time-dependences of V, including various regimes for switching on the perturbation. 

A very interesting result. The functions 1) and 2) may be identified with the ipD and ipL funetions for the D and L enantiomers (cf. p. 68) or, with the wave functions Is, centred on the two nuclei in the Hj molecule. As one can see from the last two equations, the two wave functions oscillate, transforming one to the oscillations 

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Let us consider first the two-state model of non-mteracting spin-i particles in a magnetic field. For a system... 

Measuring Protein Sta.bihty, Protein stabihty is usually measured quantitatively as the difference in free energy between the folded and unfolded states of the protein. These states are most commonly measured using spectroscopic techniques, such as circular dichroic spectroscopy, fluorescence (generally tryptophan fluorescence) spectroscopy, nmr spectroscopy, and absorbance spectroscopy (10). For most monomeric proteins, the two-state model of protein folding can be invoked. This model states that under equihbrium conditions, the vast majority of the protein molecules in a solution exist in either the folded (native) or unfolded (denatured) state. Any kinetic intermediates that might exist on the pathway between folded and unfolded states do not accumulate to any significant extent under equihbrium conditions (39). In other words, under any set of solution conditions, at equihbrium the entire population of protein molecules can be accounted for by the mole fraction of denatured protein, and the mole fraction of native protein,, ie. 

FIG. 4 Apparent mole fraction (x) water in continuous phase of brine, decane, and AOT microemulsion system derived from the water self-diffusion data of Fig. 3 using the two-state model of Eq. (1). 

Independent self-diffusion measurements [38] of molecularly dispersed water in decane over the 8-50°C interval were used, in conjunction with the self-diffusion data of Fig. 6, to calculate the apparent mole fraction of water in the pseudocontinuous phase from the two-state model of Eq. (1). In these calculations, the micellar diffusion coefficient, D ic, was approximated by the measured self-dilfusion coefficient for AOT below 28°C, and by the linear extrapolation of these AOT data above 28°C. This apparent mole fraction x was then used to graphically derive the anomalous mole fraction x of water in the pseudocontinuous phase. These mole fractions were then used to calculate values for... 

A quantitative analysis of these self-diffusion data according to the two-state model of Eq. (1) to generate the order parameter of Eq. (2) is straightforward. was found to be... 

Equation (10.6) for the mobility in the two-state model implicitly assumes that the electron lifetime in the quasi-free state is much greater than the velocity relaxation (or autocorrelation) time, so that a stationary drift velocity can occur in the quasi-free state in the presence of an external field. This point was first raised by Schmidt (1977), but no modification of the two-state model was proposed until recently. Mozumder (1993) introduced the quasi-ballistic model to correct for the competition between trapping and velocity randomization in the quasi-free state. 

Stelea SD, Pancoska P, Benight AS, et al. Thermal unfolding of ribonuclease A in phosphate at neutral pH deviations from the two-state model. Protein Sci. 2001 10 970-978. 

X-ray powder diffractometry is widely used to determine the degree of crystallinity of pharmaceuticals. X-ray diffractometric methods were originally developed for determining the degree of crystallinity of polymers. Many polymers exhibit properties associated with both crystalline (e.g., evolution of latent heat on cooling from the melt) and noncrystalline (e.g., diffuse x-ray pattern) materials. This behavior can be explained by the two-state model, according to which polymeric materials consist of small but perfect crystalline regions (crystallites) that are embedded within a continuous matrix [25]. The x-ray methods implicitly assume the two-state model of crystallinity. 

Figure 5.12 (a) The p-T phase diagram of Si. The melting lines for the low-pressure polymorph of Si and the liquid-liquid phase transition are calculated by using the two-state model and the parameters given in Table 5.2. (b) Iso-concentration lines for species B in the p-T plane, (c)) The fraction of species B as a function of temperature at constant pressure p = 2 GPa. 

In the two-state model [20,21] the two different species interact and the interaction can be expressed using the regular solution model. Thus the Gibbs energy of the liquid is... 

Figure 4.7 shows the best fits to the experimental data using Eq. (4.11). Although the data are fit within experimental error, the two-state model is certainly just an approximation. More complex distributions of sites with different quenching constants could fit the data. The success of the two-state model is not surprising given the well-known ability of two exponentials to accurately mimic complex decay curves (see above). Further, r data indicate that a more complex model is needed for a full description. 

According to the two-state model, the spectrum of the relaxed state has a mean frequency vR and is shifted relative to the spectrum of the initial state, which has a mean frequency vF. If relaxation does not occur during the process of emission (xR xF), the mean frequency of the fluorescence... 

Figure 2.5. Energy level diagram (top) and spectra (bottom) illustrating the two-state model of relaxation. The energy of the absorbed quantum is Av , and the energies of the emitted quanta are hvfl (unrelaxed) and hvF (relaxed). The fluorescence spectrum of the unrelaxed state (solid curve) is shifted relative to the absorption spectrum (dotted curve) due to the Stokes shift. The emission intensity from the unrelaxed state decreases and that from the relaxed state (dashed curve) increases as a result of relaxation.

The two-state model was used to test whether characteristics of the low-temperature cryosolvent cause the equilibrium constant for complex formation, K(T), to fall precipitously as the temperature is lowered through T ij. In this case, the slow phase that appears below 250 K would correspond to un-complexed ZnCcP. This interpretation fails because within the transition range the fraction, f(T), is unaffected by a ten-fold reduction in the ratio, R = [Cc]/[CcP], whereas use of K(T) calculated from f(T) would predict a larger shift of f(T). Alternatively, the two-state model would apply if a low-temperature form of the complex were created by a change in ligation of either ZnP or FeP. 

Fe ". In the two-state model, the electron transfer is viewed as a quantum transition between two localized states V, - and Pf. In IF,-, the ion with charge <7/ is at equilibrium with the interfacial water molecules, and the electron is in the metal. In the metal has lost one electron, and the ion with charge q/ is at equilibrium with the interfacial water. The total Hamiltonian of the system H, including all nuclear and electronic degrees of freedom, is not diagonal in the basis ( , , Pf), and so if the system is prepared in the state P, it will evolve in time according to ... 

This expression shows that the avaage occupancy of the ion s orbital takes on values between 0 and 1, depending on the solvent shift of the ion s energy relative to the Fermi level of the metal. For example, if A 0, we get (/ /> = 0, and the electron is on the metal, as we find in the case of the two-state model. 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

One can easily adjust the values of the dielectric constants D(, and Dj to obtain the experimental values of W, as in Table 4.4. With a choice of = 19.6 and Dj. = 51.0 for water, and D. = 12.5 and Dj. = 31.8 for 50% water-ethanol, we obtain the experimental values of W. We now compute the total correlation function for the two-state model for succinic acid. Here the correlation cannot be computed as an average correlation of the two configurations (see Section 4.5). The total correlation of the equilibrated two-state model is... 

The two-state model was implemented by treating the two conformers as a pair of non-interacting molecules contributing to the spin relaxation properties in proportion to their Boltzmann factors. The overall energy to be minimized for the two-state calculation is defined as ... 

We now modify the 2D continuum equations of step motion, Eqs. (7) and (8), in order to study some aspects of the dynamics of faceting. We assume the system is in the nucleation regime where the critical width Wc is much larger than the average step spacing In the simplest approximation discussed here, we incorporate the physics of the two state critical width model into the definition of the effective interaction term V(w) in Eq. (2), which in turn modifies the step chemical potential terms in Eqs. (7) and (8). Again we set V(w) = w/ l/w) as in Eq. (4) but now we use the /from Eq. (10) that takes account of reconstruction if a terrace is sufficiently wide. Note that this use of the two state model to describe an individual terrace with width w is more accurate than is the use of Eq. (10) to describe the properties of a macroscopic surface with average slope s = Mw. 

Studies of the effect of pressure on /iq for nonpolar liquids provided support for the two-state model. Pressure affects the position of the equilibrium [Eq. (23)] because of the volume change associated with trapping of the electron, A Ftr- These volume changes were deduced from changes in /td with pressure. For -alkanes [157] as well as some al-kenes [158], the mobility decreases with pressure, as shown in Fig. 11 for -hexane and 1-pentene. 

For certain liquids like cyclohexene [158], o-xylene, and m-xylene [159], the mobility increases with increasing pressure (see Fig. 11). These results provided the key to understand the two-state model of electron transport. In terms of the model, AFtr is positive for example, for o-xylene, AFtr is +21 cm /mol. Since electrostriction can only contribute a negative term, it follows that there must be a positive volume term which is the cavity volume, Fcav(e). The observed volume changes, AFtr, are the volume changes for reaction (23). These can be identified with the partial molar volume, V, of the trapped electron since the partial molar volume of the quasi-free electron, which does not perturb the liquid, is assumed to be zero. Then the partial molar volume is taken to be the sum of two terms, the cavity volume and the volume of electrostriction of the trapped electron ... 

Thus considerable support exists to support the two-state model of electron transport. The magnitude of the mobility is dependent on many factors including Fq, qf, AGsoin(e), temperature, pressure, and other factors. Presumably, differences in these factors can... 

In contrast to the assumption made in the classical occupation theory, the agonist in the two-state model does not activate the receptor but shifts the equilibrium toward the R form. This explains why the number of occupied receptors does not equal the number of activated receptors. 

Within the two-state model, the d-a system can be described in terms of two adiabatic states ij/i and ij/i with energies Ei and E2. An orthogonal transformation... 

Finally, we mention a simplified FCD (SFCD) expression for the two-state model, where Eq. 15 is reduced to [41]... 

In this section we will consider some examples of the electronic coupling between neighboring Wats on-Crick pairs, calculated in the two-state model. 

In addition, several important questions arise when one discusses large DNA fragments consisting of several base pairs. Even cases where the whole system of interest can be treated by a sufficiently accurate quantum chemical method (often this is impossible), estimates of the electronic coupling within the two-state model may lead to inaccurate results, as demonstrated for tri-mer duplexes [41]. 

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