Activation free-energy

Similarly, changes must take place in the outer solvation shell diirmg electron transfer, all of which implies that the solvation shells themselves inliibit electron transfer. This inliibition by the surrounding solvent molecules in the iimer and outer solvation shells can be characterized by an activation free energy AG. ... 

A3.8.2 THE ACTIVATION FREE ENERGY AND CONDENSED PHASE EFFECTS... 

Having separated the dynamical from equilibrium (or, more accurately, quasi-equilibrium) effects, one can readily discover the origin of the activation free energy and define the concept of the potential of mean force by analysis of the expression for the TST rate constant, k in (A3.8.3). The latter can be written as [7]... 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

A final important area is the calculation of free energies with quantum mechanical models [72] or hybrid quanmm mechanics/molecular mechanics models (QM/MM) [9]. Such models are being used to simulate enzymatic reactions and calculate activation free energies, providing unique insights into the catalytic efficiency of enzymes. They are reviewed elsewhere in this volume (see Chapter 11). 

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

If the distribution of X is assumed to be a Gaussian about the minimum of either the left- or the right-hand side, the activation free energy [Eq. (4)] becomes [82]... 

Eq. 3 is the function developed by Henry Eyring to describe equilibrium activation free energy relationships [141 ]. 

The relative magnitude of these two activation free energies determines the size and shape of the critical nucleus, and hence of the resulting crystal. If sliding diffusion is easy then extended chain crystals may form if it is hard then the thickness will be determined kinetically and will be close to lmin. The work so far has concentrated on obtaining a measure for this nucleus for different input parameters and on plotting the most likely path for its formation. The SI catastrophe does not occur because there is always a barrier against the formation of thick crystals which increases with /. 

The vertical axis is free energy, showing AGO for the net conversion of A to P, and AG, the activation free energy for each of the kinetic steps. According to Eyring s transition state theory (Chapter 7), AG is given by... 

EVALUATION OF ACTIVATION-FREE ENERGIES 3.5.1. The EVB Mapping Potential... 

FIGURE 3.10. (a) Showing the relationship between the activation free energy Ag and the reaction free energy AG0 for the X + CH3Y- XCH3 + Y system. (6) The dependence of the "linear correlation coefficient 8 = d bg /d AG0 on AG0. 

The Relationship Between Enzyme Kinetics and Apparent Activation Free Energy... 

The main lesson from the analysis given above is that the activation free energy of the reaction is strongly correlated with the stabilization of the ionic resonance structure by the protein-active site. The generality of this concept will be considered in the following chapters. 

With the valence bond structures of the exercise, we can try to estimate the effect of the enzyme just in terms of the change in the activation-free energy, correlating A A g with the change in the electrostatic energy of if/2 and i/r3 upon transfer from water to the enzyme-active site. To do this we must first analyze the energetics of the reaction in solution and this is the subject of the next exercise. 

It Is Hard to Reduce Activation Free Energies in Enzymes by Steric Strain... 

The number of particles that must be moved for complete equilibration is determined by the minimum of this expression over N. We thus determine an activation free energy profile... 

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