Modelling Enzymatic Reactions

The TS observed in a modelling process should be checked for correct geometry, calculated bond orders, and vibrational frequencies. 

---

This chapter will outline the main principles of modeling enzymatic reactions and leave to subsequent chapters the examination of specific enzymes and different catalytic factors. 

We begin our discussion with path integral quantum transition state theory (QTST) [14], which is the theoretical model that we use to model enzymatic reactions. In QTST, the exact rate constant is expressed by the QTST rate constant, qtst, multiplied by a transmission coefficient yq ... 

An interesting alternative that combines the advantages of both classical and quantum mechanics is to use hybrid QM/MM models, first introduced by Arieh Warshel for modeling enzymatic reactions [7]. Here, the chemical species at the active site are treated using high-level (and therefore expensive) QM models, which are coupled to a force field that describes the reaction environment. Hybrid models can thus take into account solvent effects in homogeneous catalysis, support structure and interface effects in heterogeneous catalysis, and enzyme structure effects in biocatalysis. 

One source of nonlinear compartmental models is processes of enzyme-catalyzed reactions that occur in living cells. In such reactions, the reactant combines with an enzyme to form an enzyme-substrate complex, which can then break down to release the product of the reaction and free enzyme or can release the substrate unchanged as well as free enzyme. Traditional compartmental analysis cannot be applied to model enzymatic reactions, but the law of mass-balance allows us to obtain a set of differential equations describing mechanisms implied in such reactions. An important feature of such reactions is that the enzyme... 

Siegbahn PEM, Borowski T. Modeling enzymatic reactions involving transition metals. Acc Chem Res. 2006 39 729-38. 

Modeling Enzymatic Reactions Using Accurate QM Models. 82... 

In this comparison, the value of electrocatalytic activity for all electrodes was normalized with respect to the rate of the model enzymatic reaction hydroquinone oxidation. The data obtained are shown in Figure 37. It is seen that a sharp reduction in the reaction rate is observed within a narrow range of distances, and r amounts to about 20 A. The shaded area corresponds to calculated data for a parabolic barrier height of 4-5 eV and other simple but by no means explicit assumptions. 

Michaelis-Menten and Briggs-Haldane models Enzymatic reactions can typically be represented as... 

So far, the most successful applications of local correlation methods have been the calculation of basic structural information geometries, frequencies, dipole moments, NMR chemical shifts and reaction energies for systems composed of over a hundred atoms and few thousand basis functions. In those cases it has been shown that 98-99 % of the correlation energy can be recovered [62]. It has also been used within QM/MM approaches to model enzymatic reactions [63]. 

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). 

Enzymatic reactions frequently undergo a phenomenon referred to as substrate inhibition. Here, the reaction rate reaches a maximum and subsequently falls as shown in Eigure 11-lb. Enzymatic reactions can also exhibit substrate activation as depicted by the sigmoidal type rate dependence in Eigure 11-lc. Biochemical reactions are limited by mass transfer where a substrate has to cross cell walls. Enzymatic reactions that depend on temperature are modeled with the Arrhenius equation. Most enzymes deactivate rapidly at temperatures of 50°C-100°C, and deactivation is an irreversible process. 

The values determined from Figure 5.23 agree well with the values calculated from the equations (Table 5.5), with an error of 3.81% for the slope and 4.65% for the intersect, respectively. The obtained experimental data were consistent with the proposed enzymatic reaction and the reaction mechanisms with uncompetitive substrate inhibition and the noncompetitive product inhibition model. 

The main point of this exercise and considerations is that you can easily examine the feasibility of the desolvation hypothesis by using well-defined thermodynamic cycles. The only nontrivial numbers are the solvation energies, which can however be estimated reliably by the LD model. Thus for example, if you like to examine whether or not an enzymatic reaction resembles the corresponding gas-phase reaction or the solution reaction you may use the relationship... 

The entropic hypothesis seems at first sight to gain strong support from experiments with model compounds of the type listed in Table 9.1. These compounds show a huge rate acceleration when the number of degrees of freedom (i.e., rotation around different bonds) is restricted. Such model compounds have been used repeatedly in attempts to estimate entropic effects in enzyme catalysis. Unfortunately, the information from the available model compounds is not directly transferable to the relevant enzymatic reaction since the observed changes in rate constant reflect interrelated factors (e.g., strain and entropy), which cannot be separated in a unique way by simple experiments. Apparently, model compounds do provide very useful means for verification and calibration of reaction-potential surfaces... 

Chapter 10 begins a more detailed treatment of heterogeneous reactors. This chapter continues the use of pseudohomogeneous models for steady-state, packed-bed reactors, but derives expressions for the reaction rate that reflect the underlying kinetics of surface-catalyzed reactions. The kinetic models are site-competition models that apply to a variety of catalytic systems, including the enzymatic reactions treated in Chapter 12. Here in Chapter 10, the example system is a solid-catalyzed gas reaction that is typical of the traditional chemical industry. A few important examples are listed here ... 

Another method that has been applied by our group to the study of enzymatic reactions is the Effective Fragment Potential (EFP) method [19]. The EFP method (developed at Mark Gordon s group at Iowa State University) allows the explicit inclusion of environment effects in quantum chemical calculations. The solvent, which may consist of discrete solvent molecules, protein fragments or other material, is treated explicitly using a model potential that incorporates electrostatics, polarization, and exchange repulsion effects. The solute, which can include some... 

APPLICATIONS TO ENZYMATIC REACTIONS 2.3.1. Active-Site and Protein Models... 

However, the active site is only a conceptual tool and the assignment of the active-site atoms is more or less arbitrary. It is not possible to know beforehand which residues and protein interactions that will turn out to be important for the studied reaction. Hybrid QM/MM methods have been used to extend the active site only models by incorporating larger parts of the protein matrix in studies of enzymatic reactions [19-22], The problem to select active-site residues appears both for active-site and QM/MM models, but in the latter, explicit effects of the surrounding protein (i.e. atoms outside the active-site selection) can at least be approximately evaluated. As this and several other contributions in this volume show, this is in many cases highly desirable. 

The next section contains the most relevant findings from ONIOM applications to enzymatic systems performed in our group. This is followed by a discussion of the important protein effects and how this information can be used to improve the modeling of enzymatic reactions. 

---