Unfolding parameters

With the exponential approximation (y 0) and the assumption that the inflow and ambient temperatures are equal, we have a stationary-state equation which links ass to tres and which involves two other unfolding parameters, 0ad and tn. Depending on the particular values of the last two parameters the (1 — ass) versus rres locus has one of five possible qualitative forms. These different patterns are shown in Fig. 7.4 as unique, single hysteresis loop, isola, mushroom, and hysteresis loop plus isola. The five corresponding regions in the 0ad-rN parameter plane are shown in Fig. 7.5. This parameter plane is divided into these regions by a straight line and a cusp, which cut each other at two points. 

Our general interest has been to find the conditions, in terms of the extra unfolding parameters p, q, r, etc., at which the qualitative nature of the stationary-state locus changes (e.g. the appearance or disappearance of a hysteresis loop or an isola). In some cases we have been able to make use of special techniques such as factorization or the tangency condition. Now we seek a more widely applicable approach. This will involve the stationary-state condition F = 0 and also a series of equations obtained by differentiation of this expression with respect to the variable x and the parameter tres. 

Applying condition (7.55), we have three equalities. Two of these may be used to eliminate x and rres, the third then gives a relationship between the unfolding parameters k2 and 0O (i.e. it gives an equation for the boundaries in the 0o k2 plane at which the isola and mushroom transitions occur). In this way we find... 

This version of the model, from 6.5, has three unfolding parameters k2, P0, and ku. If x again represents the stationary-state extent of conversion, 1 — ass, then the stationary-state condition (eqn (6.74)) can be written as... 

We now turn to the non-isothermal reaction system in a non-adiabatic CSTR, as studied in 7.2.4—6. We begin with the simplified model with exponential approximation to the Arrhenius law, and to systems for which the inflow and ambient temperatures are the same (y = 0 and gc = 0), This system has two unfolding parameters gad and rN. The stationary-state equation and its various derivatives are... 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

If we allow for the possibility of separate inflow and heat-bath temperatures, the system has four unfolding parameters, with 0C augmenting the list. The full stationary-state equation is... 

Using the singularity analysis again, the condition for the appearance or disappearance of a hysteresis loop is given in terms of the two unfolding parameters K and k2 as... 

Eigure 3.5 presents the dependence of A.S ° on temperature for chymotryp-sinogen denaturation at pH 3. A positive A.S ° indicates that the protein solution has become more disordered as the protein unfolds. Comparison of the value of 1.62 kj/mol K with the values of A.S ° in Table 3.1 shows that the present value (for chymotrypsinogen at 54.5°C) is quite large. The physical significance of the thermodynamic parameters for the unfolding of chymotrypsinogen becomes clear in the next section. 

The immobilization procedure may alter the behavior of the enzyme (compared to its behavior in homogeneous solution). For example, the apparent parameters of an enzyme-catalyzed reaction (optimum temperature or pH, maximum velocity, etc.) may all be changed when an enzyme is immobilized. Improved stability may also accrue from the minimization of enzyme unfolding associated with the immobilization step. Overall, careful engineering of the enzyme microenvironment (on the surface) can be used to greatly enhance the sensor performance. More information on enzyme immobilization schemes can be found in several reviews (7,8). 

Then, in this two-term unfolding model remains to define this exponent 2q, since all other quantities and especially the r-radius are either given, or evaluated from the thermodynamic equilibrium relations. Then, in this model the 2q-exponent is the characteristic parameter defining the quality of adhesion and therefore it may be called the adhesion coefficient. This exponent depends solely on the ratios of the main-phase moduli (Ef/Em), as well as on the ratio of the radii of the fiber and the mesophase. 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

---