Selection of a reaction coordinate

From the stationary points found along the reaction path for mechanism III we can select an appropriate reaction coordinate. We will use a geometrical coordinate which is a magnitude easy to define. However, there exist some works using an energetic or solvent coordinate[114]. A recent comparative study by Gao et. al. [260] demonstrates that calculating a PMF with the adequate sampling the results using an energetic or a geometric coordinate are equivalent.

The reaction coordinate must be a combination of the main geometrical parameters that participate in the reaction, namely, the distance between heavy atoms and transferred hydrogens r$_{NH}$, r$_{HC}$, r$_{CH}$, r$_{NH}$ as indicated in the following graphic.

In addition we can define $r_{NC}$ and $r_{CN}$ as the distances between the three heavy atoms that participate in the concerted step, that is, $N_{Lys166}-C$ and $C-N_{His297}$

These distances mainly describe the double proton transfer. An additional parameter for the configuration inversion could be the solid angle that indicates the sp3 hybridization, or as an alternative, the distance between the stereogenic carbon and the plane defined by its three bound substituents.

In table 4.1 the evolution of geometrical parameters during the reaction from S reactant to the R product is shown.

Table 4.1: Relevant geometrical parameters corresponding to the stationary points that described mechanism III found in the previous chapters

	r$_{NH}$	r$_{HC}$	r$_{CH}$	r$_{HN}$	r$_{NC}$	r$_{CN}$	r$_\alpha$	$\Delta$E
S	1.804	1.158	2.927	.995	2.884	3.844	0.4217	0.0
Sts	1.194	1.522	2.865	.996	2.645	3.765	0.4127	13.81
Is	1.151	1.570	2.845	.995	2.644	3.750	0.3984	13.79
TS	1.011	2.258	2.093	1.013	3.085	3.077	-0.0576	19.50
Ir	1.005	2.612	1.607	1.117	3.435	2.716	-0.3064	16.45
Rts	1.004	2.736	1.509	1.200	3.566	2.706	-0.3116	16.75
R	1.004	2.868	1.165	1.772	3.708	2.926	-0.3501	4.63

There is not a unique combination of parameters capable to describe the whole process. In table 4.2 we show the most intuitive combinations R$_{xx}$ for an appropriate reaction coordinate.

Table 4.2: Possible combinations of geometrical parameters to be used as distinguished reaction coordinate in the PMF calculation

	R$_{NHC}$	R$_{CHN}$	R $_{NHC-CHN}$	R$_{HCH}$	R$_{NCN}$
def.	r$_{HC}$-r$_{NH}$	r$_{HN}$-r$_{CH}$	r$_{HC}$-r$_{NH}$ + r$_{HN}$-r$_{CH}$	r$_{HC}$-r$_{CH}$	r$_{NC}$-r$_{CN}$
S	-0.646	-1.932	-2.578	-1.769	-0.960
Sts	0.328	-1.869	-1.541	-1.343	-1.120
Is	0.419	-1.850	-1.431	-1.275	-1.106
TS	1.247	-1.080	0.167	0.165	0.008
Ir	1.607	-0.490	1.117	1.005	0.719
Rts	1.732	-0.309	1.423	1.227	0.860
R	1.864	0.607	2.471	1.703	0.782

While R$_{NHC}$ and R$_{CHN}$ would be the most appropriate reaction coordinate for the proton transfer step between substrate and Lys166 and His297 respectively, we already see that they only describe partially the evolution from S to R structures.

Finally, a special computational care must be adopted when reaction coordinate Rc includes a distance from hydrogen of Lys166. Because then, the three hydrogens of amino group may act as the proton to be transferred. In consequence, although CHARMM package builds the reaction coordinate from the input atom labels, Lys166 during a MD may rotate and the labile hydrogen may change. To prevent from scanning a bad coordinate, at every step in MD we must check for the most appropriate hydrogen to be transferred, in this case, the closest hydrogen to the alpha carbon.

In what follows several reaction coordinates are employed in the mono-dimensional PMF calculation.