Two-state models

The two-state model (TSM) provides a very basic description of quantum transitions in condensed-phase media. It limits the manifold of the electronic states of the donor-acceptor complex to only two states participating in the transition. In this section, the TSM will be explored analytically in order to reveal several important properties of ET and CT reactions. The gas-phase Hamiltonian of the TSM reads 

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set 1), bb in which the Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle / 

One such rotation is usually singled out. A unitary transformation ]), (j)i,(t)2 diagonalizing the Hamiltonian matrix 

When the donor-acceptor complex is placed in a solvent, its Hamiltonian changes due to the solute-solvent interaction 

the dot product of two calligraphic letters stands for an integral over the solvent volume V 

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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... 

Now, we discuss how the geometric phase is related to the mixing angle in this two-state model. We begin by writing Eq. (A.ll) as the gauge transformation... 

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. 

Two-state model, a model of proteins that coexists in two states controlled by an equilibrium constant. Molecules with selective affinity for one of the states will produce a bias in that state upon binding to the system. Two-state theory was conceived to describe the function of ion channels but also has relevance to receptors (see Chapter 3.7). 

The Coleman-Fox two state model describes the situation where there is restricted rotation about the bond to the preceding unit (Scheme 4.3). If this is slow with respect to the rate of addition, then at least two conformations of the propagating radical need to be considered each of which may react independently with monomer. The rale constants associated with the conformational equilibrium and two values of Pirn) are required to characterize the process. 

The temperature coefficient of the potential of zero charge has often been suggested to indicate the orientation of solvent molecules at the met-al/solution interface. However, this view is based only on the response of a simple two-state model for the interfacial solvent, and on neglecting any contribution from the electronic entropy.76,77 This is in fact not the case. The temperature coefficient of 0in many instances is negative and of the... 

A controversy exists over the interpretation of such a correlation. According to the simple two-state model for water at interfaces, the higher the preferential orientation of one of the states, the higher the value of BEa=Q/BT. If the preferentially oriented state is that with the negative end of the dipole down to the surface, the temperature coefficient of Ev is positive (and vice versa). Thus, in a simple picture, the more positive BEa=0/BTt the higher the orientation of water, i.e., the higher the hydro-philicity of the surface. On this basis, Silva et al.446 have proposed the... 

In the simple two-state model shown in equation (A3.1) the receptor can only exist in either free or bound states and so... 

A simple two-state model for the observed water proton self-diffusion may be put forward in the form... 

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... 

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. 

Gougousi et al. first attempted to explain their findings by assuming the presence of Hj ions in two vibrational states (v = 0 and v, = 1) with two different recombination coefficients, and quenching of the v, = 1 ions by H2. This two-state model did not produce a consistent, quantitative fit to the data and this interpretation was abandoned. 

Fig. 8. Dependence of (A) corrected diffusion coefficient (D), (B) steady-state fluorescence intensity, and (C) corrected number of particles in the observation volume (N) of Alexa488-coupled IFABP with urea concentration. The diffusion coefficient and number of particles data shown here are corrected for the effect of viscosity and refractive indices of the urea solutions as described in text. For steady-state fluorescence data the protein was excited at 488 nm using a PTI Alphascan fluorometer (Photon Technology International, South Brunswick, New Jersey). Emission spectra at different urea concentrations were recorded between 500 and 600 nm. A baseline control containing only buffer was subtracted from each spectrum. The area of the corrected spectrum was then plotted against denaturant concentrations to obtain the unfolding transition of the protein. Urea data monitored by steady-state fluorescence were fitted to a simple two-state model. Other experimental conditions are the same as in Figure 6.

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) ... 

The results from EFISHG or HRS experiments can be corrected for resonance enhancement by using a simple theoretical two-state model (TSM, Equations (3) and (4)),35,36 which is reasonably valid for dipolar molecules in which f3 is primarily associated with a single ICT excitation ... 

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... 

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