Free-energy profile

PEP theory has also been applied to modelling the free energy profiles of reactions in solution. An important example is the solvent effect on the SN2 reaction... 

Figure A2.3.21 Free energy profile of the SN2 reaction Cl +CH2CI— [Cl-CHg-Cl]— CICH +Cl in the gas phase, dimethyl fonnamide and in water (from [93]).

The free energy profile or potential of mean force along a conformational coordinate may be defined as... 

Fig. 11.37 Free energy profile for the nucleophilic attack of water on CO2 (a) in aqueous solution and (b) in the enzyme carbonic anhydrase. (Graphs redrawn from Aqvist J, M Fothergill and A Warshel 1993. Computer Simulai of the COj/HCOf Interconversion Step in Human Carbonic Anhydrase I. Journal of the American Chemical Society 115 631-635.)...

Fig. 1. Free-energy profile for a kinetic resolution depicted by equation 1 that follows Michaelis-Menten kinetics.

Application of Eq. (30) corrects the free energies of the endpoints but not those of the intermediate conformations. Therefore, the above approach yields a free energy profile between qp and q-g, that is altered by the restraint(s). In particular, the barrier height is not that of the namral, unrestrained system. It is possible to correct the probability distributions P,. observed all along the pathway (with restraints) to obtain those of the unrestrained system [8,40]. Erom the relation P(q)Z, = P,(q)Z, cxp(UJkT) and Eqs. (6)-(8), one obtains... 

Eor many problems, the ideal umbrella potential would be one that completely flattens the free energy profile along q, i.e., UXq) = W(q). Such a potential cannot be determined in advance. However, iterative approaches exist that are known as adaptive... 

The umbrella sampling and eonstrained dynamies teehniques are suffi-eiently general to investigate the free energy profile not only as a funetion of... 

Outer sphere electron transfer (e.g., [11-19,107,160-162]), ion transfer [10,109,163,164] and proton transfer [165] are among the reactions near electrodes and the hquid/liquid interface which have been studied by computer simulation. Much of this work has been reviewed recently [64,111,125,126] and will not be repeated here. All studies involve the calculation of a free energy profile as a function of a spatial or a collective solvent coordinate. 

D Free Energy Profile for Cation Passage Through the Gramicidin Channel 188... 

Fig. 8. A. Free energy profile for Na+ movement through the malonyl Gramicidin channel. Calculated using Eyring rate theory and the locations of binding sites in the channel.

B. Schematic representation of the channel aligned with the free energy profile above and indicating the direction of the field. Reproduced with permission from Ref. 45)... 

Calculation of Single Channel Currents and the Equivalent Free Energy Profile... 

FIGURE 3.5. The actual free-energy profile for the ground-state surface as a function of the energy gap As. The calculations are done for the CF + CH3C1- C1CH3 + CP exchange reaction (Ref. 11). 

FIGURE 5.3. A schematic free-energy profile for a reference reaction in a solvent cage. 

FIGURE 8.4. Calculated free-energy profiles tor the reference reaction in solution, Ags, and for the enzyme reaction, Agp. 

FIGURE 8.8. Calculated free-energy profile for the reaction of carbonic anhydrase. g2(a) and g2(b) designate the states where the proton acceptors are water and histidine respectively. 

FIGURE 6.1 Free energy profile of a reaction without an intermediate where the products have a lower free energy than the reactants. 

FIGURE 6.2 (a) Free energy profile for a reaction with an intermediate. AG and AG are the free energy of activation for the first and second stages respectively, (b) Free energy profile for a reaction with an intermediate in which the first peak is higher than the second. 

Figure 6.3 shows a free energy profile for a reaction in which B is thermodynamically more stable than C (lower AG), but C is formed faster (lower AG ). If neither reaction is reversible, C will be formed in larger amount because it is formed faster. The product is said to be kinetically controlled. However, if the reactions are reversible, this will not necessarily be the case. If such a process is stopped well before the equilibrium has been established, the reaction will be kinetically controlled since more of the faster-formed product will be present. However, if the reaction is permitted to approach equilibrium, the predominant or even exclusive product will be B. Under these conditions, the C that is first formed reverts to A, while the more stable B does so much less. We say the product is thermodynamically controlled.Of course. Figure 6.3 does not describe all reactions in which a Compound A can give two different products. In many cases, the more stable product is also the one that is formed faster. In such cases the product of kinetic control is also the product of thermodynamic control. 

FIGURE 6.3 Free energy profile illustrating kinetic versus thermodynamic control of product. The starting compound (A) can react to give either B or C. 

FIGURE 10.1 Free energy profile for the gas-phase (solid line) and aqueous soltuion (dashed line) Sn2 reaction between CH3CI and Cl , from MO calculations. ... 

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