Two-state mechanism

Analysis of the poly(methyl methacrylate) sequences obtained by anionic polymerization was undertaken at the tetrad level in terms of two different schemes (10) one, a second-order Markov distribution (with four independent conditional probabilities, Pmmr Pmrr, Pmr Prrr) (44), the other, a two-state mechanism proposed by Coleman and Fox (122). In this latter scheme one supposes that the chain end may exist in two (or more) different states, depending on the different solvation of the ion pair, each state exerting a specific stereochemical control. A dynamic equilibrium exists between the different states so that the growing chain shows the effects of one or the other mechanism in successive segments. The deviation of the experimental data from the distribution calculated using either model is, however, very small, below experimental error, and, therefore, it is not possible to make a choice between the two models on the basis of statistical criteria only. 

Hen egg-white lysozyme, lyophilized from aqueous solutions of different pH from pH 2.5 to 10.0 and then dissolved in water and in anhydrous glycerol, exhibits a cooperative conformational transition in both solvents occurring between 10 and 100°C (Burova, 2000). The thermal transition in glycerol is reversible and equilibrium follows the classical two-state mechanism. The transition enthalpies AHm in glycerol are substantially lower than in water, while transition temperatures Tm are similar to values in water, but follow similar pH dependences. The transition heat capacity increment ACp in glycerol does not depend on the pH and is 1.25 0.31 kj (mol K) 1 compared to 6.72 0.23 kj (mol K)-1 in water. Thermodynamic analysis of the calorimetric data reveals that the stability of the folded conformation of lysozyme in glycerol is similar to that in water at 20-80°C but exceeds it at lower and higher temperatures. 

At the time that this simple two-state mechanism was conceived—and, indeed, in most attempts to model unknown structures—the proteins were pictured as smooth and symmetrical. In keeping with this image, the motions associated with both regulation and contraction were pictured by many as being simple and machine-like (but see an alternative view in Squire, 1975). 

It has been shown that under standard conditions most singledomain globular proteins exhibit a folding/unfolding behavior consistent with the two-state mechanism (Freire and Biltonen, 1978a Privalov, 1979). From a statistical thermodynamic standpoint the implication is that the population of partially folded intermediate states is negligible and the partition function reduces to two terms ... 

Oxygen atom from Cpd I is inserted into the C-H bond of saturated hydrocarbons (Scheme la) by means of hydrogen atom abstraction followed by recombination of the transient hydroxyl with the carbon radical [the so-called oxygen rebound mechanism proposed by Groves in 1976 (8, 10)]. Another possibility can be the concerted oxygen insertion into the C-H bond. Both pathways are rationalized by the two-state mechanism developed by Shaik et al. (6, 9), which describes different reactivities... 

Insertion of oxygen atom from Cpd I into the carbon-carbon double bond with formation of epoxide (Scheme Ic) reveals features characteristic for a concerted process, although formation of radical intermediates is possible in many cases. A unified description of this alternative is also provided by the two-state mechanism of catalysis by Cpd I (see the section on Hydroxylation of hydrocarbons). Essentially, the concerted oxygen insertion represents a low-spin reaction surface, whereas the distinct radical intermediate is formed on the high-spin reaction pathway. In the latter case, the carbon radical may attack the nearby heme nitrogen and modify the heme covalently. This reaction is also an important inactivation pathway of cytochromes P450 during oxidative transformations of terminal double and triple bonds. 

The series mechanism of protein folding may or may not involve the presence of stable and detectable intermediates. The following two-state mechanism is involved during the folding of small proteins ... 

Experiments for investigating the lamellar-Hn transition kinetics have been performed, for example, on DOPE dispersions. The F,p-phase diagram of DOPE in excess water is depicted in Fig. 21. Figures 22, 23 show the diffraction patterns and lattice parameters at 20 °C after a pressure jump from 300 to 110 bar. Clearly, the (001) reflection of the L phase and the (10) reflection of the developing Hu phase can be identified. In this case, a two-state mechanism is observed. Interestingly, we find that successive pressure jumps lead to an acceleration of the phase transition kinetics. The half transit time decays from 8.5 s for the first pressure jump to... 

When only two kinds of ion pairs are in the system (dissociation being suppressed by addition of counter ions), the polymerization proceeds via a two-state mechanism. In this case the nonuniformity is composed of two terms ... 

The mechanism given by eqn.(22) implicitly involves ihat the rates of hydrogen bond formation and breaking are independent of the position from which the acceptor and donor sites are delivered and consequently are indepent of the degree of polymerization. The two state mechanism thus corresponds to the approximation given by eqn.(3). In terms of that approximation eqns.(24) and (25) rewrite as... 

Eqn.(28) predicts linearity of (1/t) y the overall concentration of NMA. It is remarkable that the predicted concentration dependence of the relaxation time is identical with the one predicted from the far simpler monomer/dimer mechanism. Fig. 1 shows the concentration dependence of (I/t) obtained from the ultrasonic experiments and it appears that the concentration dependence predicted by the two state mechanism agrees with the data. 

Fig. 16. Fluorescence spectra of apoCopC in the absence or presence of urea, in 20 mM PBS and 0.1 M NaCl, pH 6.0. The concentration of urea / M is 1 0 2 5 3 9.4, respectively. 4 is the fitted curve according to two-state mechanism.

Fig. 17. GdnHCl-induced unfolding of CopC by monitoring the fluorescence 400/320 nm ratio for apoCopC (o), Cu2+-CopC ( ), CopC-Ag+ (A), and Cu2+-CopC-Ag/ ( ) forms at pH 7.4, 25°C. The solid lines are fitting curves in terms of two-state mechanism for apoCopC (o) and CopC-Ag+ (A), of three-state for Cu2+-CopC ( ) and Cu2+-CopC-Ag ( ) from eq (2) and eq (8), respectively. Inset Dependence of free energy change of Cu2+-CopC ( ) on GdnHCl concentration.

Litvinienko and Mueller. This treatment involves several instances two-state mechanism with unimolecular isomerization, bimolecular exchange, degenerative transfer of two chain ends of different activities, and aggregation of two chain ends of different activities. 

Figini and others " have shown that, for a two-state mechanism, a slow exchange between various active (or between active and dormant) spedes leads to a broadening of the MWD as given in eqn [10] ... 

Although there are exceptions, many of the smaller proteins studied to date fold by a two-state mechanism with a single rate-limiting transition state. The transition state is a crucial but unstable stage along a folding pathway traversal back to the native state has equal probability to that of the system progressing to the unfolded state. Transition states are often a less compact version... 

In the present discussion, only the studies at equilibrium are described. From the data reported in the literature, one can discuss the validity of the analysis according to a two-state mechanism for several examples, and for other examples the deviation from a two-state behavior. The kinetic aspect of the problem is analyzed in Chapter 7. Specific methods and particular... 

The classical method of analysis assumes a two-state mechanism... 

As pointed out by Tanford (1968) and by Pace (1975), the analysis in term of a two-state mechanism can be useful even when the transition is known to deviate significantly from a two-state behavior. In this case, Pace (1975) recommended to term X pp and AG pp, the equilibrium constant and the free energy variation of unfolding, respectively. 

Tanford (1968) considered the situation where unfolding deviates from a two-state mechanism and took into account the presence of intermediary species, ATj, each being characterized by the property being the fraction of intermediates present at equilibrium. Thus y becomes at any point ... 

It has been noted (Pace, 1975) that urea- and GuHCl-induced transitions are generally closer to a two-state mechanism than pH- or temperature-induced denaturation. 

The degree of departure from a two-state mechanism varies widely with the proteins, and for a given protein varies with the denaturant. It also depends on the other experimental conditions. In many cases, only kinetic studies have revealed the occurrence of intermediates (see Chapter 7). When most of the equilibrium studies are such that conditions (1), (2), and (3), or even only (1) and (2), are satisfied, it may be reasonably concluded that the intermediate states, if they do exist, are not significantly populated, and that the equations of a two-state model may be used in a first approximation. Useful information can be obtained by such analysis. Furthermore, when conditions (3), (4), and (5) are fulfilled, the validity of a two-state mechanism is demonstrated. 

It appears from different research reports (Tanford, 1970 Pace, 1975) that, for many proteins, reversible transitions from the unfolded to the folded state have characteristics of a two-state process, without detectable intermediates at equilibrium, even when the existence of intermediates is supported by other experimental arguments (kinetics for example, see Chapters 7, 8 and 9). Urea and GuHCl denaturation is generally closer to a a two-state mechanism than pH or thermal unfolding. 

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