Free-energy profiles, computation

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.)...

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. 

With the characterized mechanism, the next key question is the origin of its catalytic power. A prerequisite for this investigation is to reliably compute free energy barriers for both enzyme and solution reactions. By employing on-the-fly Born-Oppenheimer molecular dynamics simulations with the ab initio QM/MM approach and the umbrella sampling method, we have determined free energy profiles for the methyl-transfer reaction catalyzed by the histone lysine methyltransferase SET7/9... 

Figure 4.13 shows the free energy profile as a function of the helix-helix distance. Equation (4.47) allows the computation of the contributions to the profile by the different intermolecular potentials. The helix-helix and helix-solvent interactions were considered. The helix-helix van der Waals potential shows a significant minimum... 

The difficulty of such treatments are that they do not provide expressions for the rate of die ET reactions at electrodes which can be compared with experiments. They involve complicated computer simulations to determine the free energy profile. Such simulations generally use adjustable parameters to make the results fit experiments. Conversely, these treatments include both short- and long-range ion-solvent interactions and the interaction of the ion and the solvent with the metal electrode at a molecular level. 

To evaluate solvent effeets, statistieal meehanical Monte Carlo simulations have been carried out. An important quantity to be computed is the potential of mean force, or free energy profile, as a funetion of the reaction coordinate, X, for a chemical reaction in solution using free energy perturbation method. (44) A straightforward approaeh is to determine free energy differences for incremental changes of certain geometrieal variables that characteristically reflect the... 

Figure 7 shows the free energy profile as a function of the energy-gap solvent reaction coordinate, which is compared with the PMF as a function of XR. The computed w(Xs) also has a unimodal shape and the estimated free energy barrier is 16.1 kcal/mol, in good agreement with the value from Fig. 6. Thus, for the Type 4 reaction, the use of a geometrical and a solvent reaction coordinate does not affect...

Though the computed free energy profile for the 8, 2 reaction in water is unimodal, it seemed likely that there should be a solvent that could yield a nonconcerted profile, that is, one with ion-molecule or solvent-separated intermediates. A solvent with diminished anion solvating ability should reduce the energy increase upon desolvation of the nucleophile. This could at least reintroduce the ion-molecule complexes as intermediates. Though hydrocar-... 

In order to gain a quantitative understanding of the catalytic power of enzymes, it is essential to be able to calculate the free-energy profiles for enzymatic reactions and the corresponding reference solution reactions. The common prescription of obtaining potential surfaces for chemical reactions involves the use of quantum mechanical (QM) computational approaches, and such approaches have become quite effective in treating small molecules in the gas phase (e.g., Ref. 5). However, here we are interested in chemical reactions in very large systems, which cannot be explored... 

In the previous section we have outlined computational methods to accurately obtain free energy profiles as a function of an order parameter, E. On the one hand, these one-dimensional free energy profiles contain a wealth of information. On the other hand, the configuration space is of very high dimension and a one-dimensional order parameter might not be sufficient to construct... 

The dependence of the error on the simulation parameters becomes more transparent if e is expressed as an explicit function of the total simulation time. Consider in fact a free energy profile F (s) that has to be filled with Gaussians up to a given level Tmax, for example the free energy of the lowest saddle point in F (s). The total computational time needed to fill this profile can be estimated as the ratio between the volume that has to be filled and the volume of one Gaussian times tq ... 

Computing Free-Energy Profiles Using Multidimensional Potentials of Mean Force and Polynomial Quadrature Methods... 

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