Multi-scale modelling of transition metal enzymes

Transcription

1 Degree Project C in Chemistry, 15 c Multi-scale modelling of transition metal enzymes Nathalie Proos Vedin 18th June 2015 Supervisor: Marcus Lundberg Uppsala University Department of Chemistry Ångström Laboratory

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3 Summary The behaviour of enzymes is something of great interest to many science fields today. A deeper understanding of how they work can for example lead to the development of new and efficient catalysts. There are different ways to approach the problem of investigating enzymes and the procedure chosen in this thesis is the use of a theoretical method called QM/MM. The main idea of the method is to study the active site of the enzyme on a higher level of accuracy than that of the rest of the protein. This opens up for the possibility to study the reaction mechanism by which the enzyme works. However, a problem that arises with the use of theoretical methods is that it is difficult to evaluate the accuracy of the results. Thus, the objective of this thesis was to find out how to deal with this issue when performing calculations on transition metal enzymes. A protein which has recently been studied with the aid of a QM/MM method, called HPPD, was chosen and the aim was to evaluate the stability of the conclusions drawn in the published paper [1]. The factors that affected the energy values were identified and their importance to the end result was investigated. In particular, it was found that the choice of method used for the more accurate calculations on the active site affected the results considerably. Moreover, the description of the interaction between the active site and its surrounding protein and how the model for the computations were chosen were shown to be two other factors that influenced the energy results. In the end, however, the conclusions that could be drawn was that the results in the investigated study of HPPD were trustworthy.

4 Abbreviations D2 D3 DFT EE GGA HF HG HPA HPP HPPD INT1 MM NIM P PCM QM QM/MM R TS1 TS2 vdw ZPVE Grimme s DFT-D2 dispersion correction method Grimme s DFT-D3 dispersion correction method Density functional theory Electronic embedding Generalized gradient approximations Hartree-Fock Homogentisate Hydroxyphenylacetate 4-hydroxyphenylpyruvate 4-hydroxyphenylpyruvate dioxygenase The intermediate of the proposed reaction mechanism Molecular mechanics ur N -layered integrated molecular orbital and molecular mechanics The product of the proposed reaction mechanism Polarizable continuum model Quantum mechanics Combined quantum mechanics and molecular mechanics The reactant of the proposed reaction mechanism The first transition state of the proposed reaction mechanism The second transition state of the proposed reaction mechanism van der Waal Zero-point vibrational energy

5 Contents 1 Introduction Theory and computational details MM calculations QM calculations Basis set Dispersion effects The QM/MM method Interactions between the subsystems Solvation models Thermodynamical aspects Computational details Results Effects Sensitivity QM energy Protein environment Model Discussion Recommendations Choice of functional Size of QM part Model with or without water Final comments

6 1 Introduction The study of the catalytic behaviour of enzymes is of great importance to many fields of research today. Their ability to aid complex reactions are far better than that of any man-made catalysts as of yet, which is why we are interested in being able to replicate their function. The enzyme studied in this thesis is a transition metal enzyme belonging to a family of non-heme iron enzymes. Transition metal enzymes are significant in many important biochemical processes, and knowledge about their efficiency and selectivity is highly desirable in the field of biomimetics (the field concentrated on the design of systems inspired by nature) where they can serve as inspiration for new catalysts. For example, they have been found to be of great interest to research investigating the antibiotic resistance of bacteria [2] and the development of solar fuels [3]. In order for us to understand the behaviour of enzymes and how they achieve such stupendous catalytic ability we need to know by which method they work, i.e. what the reaction mechanism is. In this thesis a theoretical approach has been used in order to answer this. ne of the main advantages with using computational methods to study reactions is the possibility to study parts of the mechanism otherwise inaccessible to us due to their very short life time in reality (such as transition states). Even meta-stable intermediates pose difficulties when handling complex systems such as enzymes. It is imperative to be able to study these short-lived structures since they most often are what defines the sought reaction path [4]. Fortunately, with theoretical methods it is nearly as easy to study these more problematic parts of the reaction as it is to study the stable structures of the mechanism. By calculating and comparing the different energy barriers for possible reaction steps we can draw conclusions concerning the actual reaction [5]. Furthermore, a theoretical approach also opens up for the possibility to analyse the effects from different parts of the system separately, which for obvious reasons it not possible with the use of experimental methods. Although the computational methods have evolved noticeably over the years, studying as complex and large systems as enzymes still poses a great challenge. ne way to approach this is to study the active site, i.e. the part of the protein where the reaction occurs, separately from the rest of the enzyme. Nevertheless, since actually all parts of the enzyme, which can consist of thousands or tens of thousands of atoms, contribute to the total reaction energy one would ideally apply calculations on the complete system. The method to do so is called QM/MM, short for combined quantum mechanics and molecular mechanics, and is a hybrid method in which different parts of the protein is studied with different levels of accuracy [6]. The main idea is that the system is divided into different parts according to their relevance to the studied reaction. A schematic illustration of this can be found in Figure 1. The chemically active part is included in the QM model, which is a set consisting of relatively few atoms (around ) on which more accurate, although at the same time more computationally demanding, quantum mechanical calculations are performed. The rest of the system, i.e. the greater part of the protein, is included in the MM part where the energy is computed with a so called 6

7 MM QM Figure 1: Schematic representation of the subsystems in a QM/MM calculation. empirical force field method, based on classical calculations. The molecular mechanics calculations are a lot less expensive in terms of time and computational effort and may therefore include thousands or tens of thousands of atoms. However, they are generally not able to deal with the breaking and formation of bonds, which is why a combination of both QM and MM is used. The QM/MM method has come to be an important tool to study biocatalysts today [4]. ne of the greatest problems with applying these methods to such complex systems is that it is very difficult to know how accurate the obtained results are. In most cases there is no experimental data to compare them with and the size of the systems prevents us from making a complete QM calculation. Another aspect that one needs to keep in mind, however, is that in addition to the precision of the method, the accuracy also depends on how the model is chosen. The results may vary greatly depending on for example the size of the QM part, if there is water or other solvent molecules present or of how the two subsystems (QM and MM) interact. In order to determine what the reaction mechanism is we would ideally want calculated results to give the exact energies of each reaction barrier. This is wherein the problem of not knowing the accuracy lies: obtaining results that coincide with experimental results does not necessarily mean that the reaction mechanism they describe is the correct one, but could arise due to substantially large cancellation of errors [4]. The actual reaction path could also be another, unexplored, mechanism which would fit the experimental data even better. We can, however, use our results to compare different proposed reaction paths (e.g. suggested by experiments) and estimate which are the most probable ones. Thus, the results obtained by theoretical calculations cannot be used to find an absolute truth, but to rule out which mechanisms are not very probable descriptions of the reality [7]. The more we know about the accuracy of our results, the more valid conclusions we can draw from them. The objective of this thesis is to apply additional calculations to a previously studied system in order to evaluate the stability of the results [1]. The chosen system is an enzyme called 4-hydroxyphenylpyruvate dioxygenase, abbreviated HPPD, and it can be found in most aerobic life forms (Figure 2). In humans HPPD 7

8 is involved in the catabolism of tyrosine and a deficiency of the enzyme gives rise to a metabolic disorder called tyrosinemia type III. In plants HPPD plays a role in the process of photosynthesis, which is why inhibitors of the enzyme are used in many commercial herbicides [1]. The reason for choosing this particular system is that it is a transition metal enzyme for which the reaction mechanism has recently been determined with the aid of a QM/MM method [6]. Figure 2: The studied enzyme, 4-hydroxyphenylpyruvate dioxygenase. The atomic coordinates were obtained from reference [1] and the inset shows the atoms included in the QM part of the calculations performed. The proposed reaction mechanism for the process when HPPD catalyses the oxidation of its substrate 4-hydroxyphenylpyruvate (HPP), leading to the formation of 2,5-dihydroxyphenylacetate (homogentisate, HG) can be seen in Figure 3 [1]. It is the reaction steps following the decarboxylation and the heterolytic cleaving of the bond giving the high-spin complex HPPD Fe(IV) HPA (where HPA is an abbreviation for hydroxyphenylacetate) that is the aim of this thesis. It has been proposed [1] that the mechanism consists of electrophilic attack on the C1 carbon of the aromatic ring followed by migration of the acetic acid substituent from C1 to C6. The studied structures in this thesis are therefore the ones present in Figure 4, where the Fe(IV) HPA has been labelled R and the product HG is called P. 8

9 H H H 2 C 2 His Fe II His Glu H 2 His His Fe III Glu His His Fe II Glu H H His Fe IV His Glu HPPD Fe(IV) -HPA H His Fe II His Glu HPPD Fe(II) HG Figure 3: The conversion of HPP into HG, catalysed by HPPD. The mechanism includes oxidation, decarboxylation and heterolytic cleaving of the bond. Adapted from reference [1]. H H H C2 C1 C6 Fe IV TS1 Fe III TS2 Fe II R INT1 P Figure 4: The last reaction steps of the mechanism in Figure 3, studied in this thesis. The abbreviations are (from left to right): reactant, transition state 1, intermediate 1, transition state 2 and product. Adapted from reference [1]. Apart from the reaction mechanism depicted above, four other mechanisms were considered in the paper. These mechanisms did instead include electrophilic attack on two other carbons in the aromatic ring (C2 and C6) and the formation of an epoxide intermediate. Table 1 lists the energy values for the first transition states of all of them. Mechanism A is the mechanism shown in Figure 4 and had the lowest energy barrier. 9

10 The mechanism closest in energy to that of our mechanism (labelled A) has been termed mechanism B and the difference between the two transition states is just below 9 kcal/mol. 1 This means that if the energy barrier of mechanism A is shown to increase with 5 kcal/mol at the same time as the one in mechanism B decreases by an equal amount the latter would be an equally probable description of reality. Table 1: Energies of the first transition states of the five reaction mechanisms that were investigated in the paper [1]. Mechanism Energy of first TS [kcal/mol] A 18.0 B 26.8 C 41.4 D 29.2 E 34.7 Furthermore, the calculations of mechanism A indicates that the rate limiting reaction step is the first barrier, since the two transition states were found to be 18.0 and 14.7 kcal/mol, respectively. Knowing what the rate limiting step is becomes important when one wants to be able to influence the mechanism in some way. For instance, lowering the energy barrier of any of the other reaction steps is futile if the highest barrier stays the same. In summary, the objective of this thesis is to study the enzyme HPPD and use it as an example of how to go about the problem of evaluating the accuracy of obtained theoretical results. In order to do so in a realistic way we have limited our study to answer two quite simple, but very essential questions about this system, following below: Are the conclusions drawn in the paper regarding the reaction mechanism reasonable or could any of the other studied mechanisms be equally probable? Is it possible to determine what the rate limiting step (i.e. the highest reaction barrier) is? These are questions that need to be answered in order for anyone to be able to draw conclusions regarding this system. By varying the model and methods used we will try to evaluate which parts are affecting the results more than others and how important their contribution to the end result is. With this study we hope to show how theoretical studies can be beneficial to biochemical research and to contribute to the understanding of HPPD and QM/MM modelling in general. 1 The energies here, and also through the rest of this report, are presented in kcal/mol since this is the conventionally used energy unit within the field of computational chemistry. ne kcal/mol is equivalent to 4.18 kj/mol [8]. 10

11 2 Theory and computational details This section presents the theory behind all effects studied in this thesis. We begin by presenting the separate calculation methods (MM and QM) and then move on to the procedure of combining them. Finally, the computational details regarding the calculations performed during this project are described. 2.1 MM calculations Molecular mechanics calculations are performed with so called force field methods, which are based on classical mechanics. The ones most widely used for biological systems are called AMBER, CHARMM and PLS-AA [6] and in this thesis we have used the first of them. Due to the fact that all atoms included in the calculations are treated as point charges, force field methods are very cheap in terms of computation time. It also enables us to perform MM calculations on very large systems, such as enzymes and other macromolecules. n the other hand, the above mentioned simplification makes these types of methods unable to describe the formation and breaking of bonds in a correct way, which of course is a great disadvantage when one wants to investigate reaction mechanisms [6, 9]. Furthermore, in order to use force field methods one needs to find additional parameters that describe the system. This is something that in general can be quite difficult and even more so for transition metal systems due to the possibility for them to exist in different spin states. The energy computed by an MM method can be divided into different energy contributing parts. These are presented in the following equation: E MM = E bonded + E vdw + E Coulomb (1) where the bonded terms include bond stretching, angle bending, torsions and out-ofplane deformations and where E vdw and E Coulomb describes the van der Waals and the electronic interactions within the studied subsystem, respectively [6]. 2.2 QM calculations The problems with MM calculations can be solved by the use of quantum mechanical methods. In contrast to the force field methods, these are more accurate and able to describe exactly what happens in each reaction step. However, as a consequence to this, these calculations require considerably more time and computational effort which limits their use to quite small systems (or subsystems). In general it is good to keep in mind that a greater QM part will, while requiring more demanding calculations, most probably give results closer to reality. 11

12 There are many different ways in which these calculations can be done. ne way to calculate the energy for a system at the quantum chemical level is to directly solve the time-independent, non-relativistic Schrödinger equation: ĤΨ = EΨ (2) where Ĥ is the Hamiltonian operator, Ψ the wavefunction and E is the energy of the system. This approach, however, is quite costly, especially for large systems. Since there are other, more suitable ways to calculate the energy of transition metal systems this method will not be explained further. Another quicker and more efficient method is the so-called density functional theory, abbreviated DFT. It has been proven [10] that the ground state energy of a system can be directly obtained from its electron density, which is easier to obtain since you do not have to derive any wavefunctions. The problem, however, is that we do not know any perfect way to translate the density into the energy of the system yet. Therefore, there is a vast amount of different functionals (which is a type of function of a function) used in research today, and their performance vary greatly [11, 12]. In general almost all functionals are compatible with the QM/MM approach, although some might be better suitable than others from an accuracy perspective. Which functional one should choose depends on the type of system, i.e. which kinds of bonds and interactions that are present. They are often categorized according to their complexity on something called Jacob s ladder [13]. Table 2 lists all functionals used in this study, divided into each rung of this ladder. There are more groups of functionals than the four described next, but only the ones relevant for this thesis are presented here. The first rung is the so called generalized gradient approximations (GGA), where the density and its gradient are calculated, and the most popular functional in use is the BLYP functional. The next step contains the hybrid functionals. These use a mix of GGA and so called exact Hartree-Fock exchange (HF exchange). The amount of HF exchange for each hybrid functional is also included in Table 2. Including Hartree-Fock exchange to some extent to the DFT functionals gives a more exact, but at the same time more costly, calculation method and has been shown to give more accurate results [11, 13]. However, the optimal amount of HF exchange vary greatly depending on which system the method is applied to. In general it can be said that pure DFT favours low-spin states relative to high-spin states. This is of special interest when calculating energies of transition metal complexes and systems, since they can exist in different spin states [11, 13, 16]. The third and forth rung of the ladder presented in Table 2 are the metageneralized gradient approximations which, in addition to the density and its gradient, also take the kinetic energy density into consideration and the meta hybrid GGAs. The most popular and commonly used functional in the field of transition metal enzymes is B3LYP, which is a hybrid functional with 20 % exact HF exchange. ne of the reasons for its success is that the errors resulting from B3LYP calculations cancel each other out 12

13 Table 2: Jacob s ladder of DFT functionals, along with the amount of HF exchange in the hybrid ones [14, 15]. Functional HF exchange GGA functionals BLYP BP86 Hybrid functionals B3LYP 20% B3LYP* 15% Meta-GGAs TPSS M06-L Meta hybrid functionals TPSSh 10% M06 27% in a favourable way. This, on the other hand, makes the method difficult to improve in general, since adjusting one of the factors would erase the cancellation effect and thus lead to more inaccurate results [17]. In contrast, reported results from calculations on transition metal enzymes made with this functional have been shown to vary somewhat [13, 17, 18]. Instead, the accuracy of the results has been found to be improved by lowering the amount of exact exchange in the hybrid functional. The functional B3LYP*, with 15 % HF exchange was proved to give considerably better results on transition metal systems, and also B3LYP ± (with 10 % HF) to some extent [14, 16, 17, 18]. Nevertheless, the most common procedure is still to use the B3LYP functional for the main calculations and thereafter to use the B3LYP* functional only to evaluate the stability of the results [17]. The reason for the continued use of B3LYP is probably due to the fact that using the same calculation method (although not the best one) makes for easy comparisons [5] Basis set An approximation included in the quantum mechanical calculations considered here is the introduction and use of a basis set. Basis sets are sets of functions that can be used to describe another function, such as wavefunctions calculated in QM methods (much like the Taylor series is used to approximate polynomials). The larger the set, the better the approximation. The simplest type of basis set is a so called minimum basis set and apart from that there are different expansions in use, such as the so called double-ζ (with double the amount of basis sets) and triple-ζ basis sets [11]. 13

14 2.2.2 Dispersion effects Although considering the electron density instead of the wavefunctions certainly makes the energy calculations more efficient, many common DFT functionals (such as B3LYP) has one significant shortcoming, namely its inability to describe the van der Waals interactions of the system. These are very important in many larger chemical systems, which makes this insufficiency troublesome [18, 17, 19]. A solution to this problem is to use a so-called dispersion correction when calculating the energy of the system. This correction introduces empirical van der Waals effects into the model and the energy contribution can be read separately from the original energy results. Two commonly used dispersion correction methods are the ones developed primarily by Stefan Grimme called DFT-D2 and DFT-D3, where the latter are the latest published one with more refined properties [19]. 2.3 The QM/MM method As previously mentioned the main objective with using a QM/MM approach on biological systems, such as enzymes, is that one is able to include the whole protein into the calculation and still get satisfactory results. This is possible due to the partitioning of the system into (often two) parts, see Figure 1. Each atom in the system is assigned to either one of the subsystems, but due to the strong interactions between the two regions it is not possible to simply add their energies together. There are different ways to handle these interactions, one being to carry out the three following calculations on the system: an MM calculation on the complete system (sometimes termed the real) followed by both a QM and an MM calculation of the inner subsystem (the model). The resulting scheme, described in equation 3, is called a subtractive scheme and is the one used for the calculations in this thesis. The method used here is called NIM, which is short for our N-layered integrated molecular orbital and molecular mechanics and in our case the performed calculations involved two and three layers. E QM/MM = E MM,real + E QM,model E MM,model (3) The main advantage with this scheme is that it is relatively simple to implement and that one does not have to calculate the interaction between the subsystems explicitly [6]. An important question that is sometimes asked is whether one really has to use the QM/MM approach on enzymes, with all its accompanying complications, or if it is enough to study the QM part separately in order to get results that are trustworthy. The conclusion that can be drawn is that it depends on the studied system and its properties [20]. To evaluate the significance of the different parts of the QM/MM calculation one can study their energy contributions individually. That way it becomes clear if e.g. the MM part contributes considerably to the end result. This is easily done with a subtractive scheme, since the separate calculations are performed automatically. 14

15 2.3.1 Interactions between the subsystems As described above, the interactions between the two subsystems are very important and can be divided into covalent, van der Waals and electrostatic interactions. It is important to know how to handle these in order to get as good results as possible. For example, the covalent interactions between the subsystems becomes interesting for the cases when the boundary of the QM part cuts through a covalent bond. This is most commonly solved by the use of so called link atoms, which are added to the lose ends in the QM part in order for the description of the system to stay reasonable [6]. The electrostatic interactions between the inner and the outer regions, on the other hand, can be treated at different levels of precision. The choice of so called embedding scheme depends on computational resources and the properties of the protein environment [21]. The two most commonly used are termed mechanical and electronic embedding. In a mechanical embedding scheme the electrostatic interactions between the two subsystems are treated the same way as within the MM region (most commonly as point charges). This is a quite easy and effective method, which is mainly used since the results are easily analysed. However, the simplicity comes at a cost; for example, describing the QM region with fixed charges means that they would have to be reparametrized if those charges were to change during the reaction. By using an electronic embedding (EE) scheme a lot of the problems with the mechanical one is fixed. The energy of the QM region is calculated in the presence of the MM charges, and thus, the inner region can adapt to changes in the protein environment. Evidently, this embedding scheme requires more computational time and effort, but it is often considered necessary in order to obtain reliable results [6, 22]. This is especially important if the surrounding protein is polar [21] Solvation models Another thing worth taking into consideration is the environment of the complete system. There is a possibility to add a third layer to the QM/MM calculation that includes the solvent (in this case an aqueous solution) in which the enzyme exists [9]. The concept is called a continuum approach and one of the most frequently used models is the so called Polarizable Continuum Model, in our case NIM-PCM [23]. The atoms of the solute molecule (i.e. the protein) is put inside a cavity having the same shape as the original molecule, but with a larger radii for each atom. This is done in order to create a surface without openings with which the solvent can interact. The solvent is then treated as a continuum of charges on the surface, rather than separate atoms or molecules. This simplicity makes it computationally possible to add this new solvent layer consisting of such a vast amount of atoms [24]. 15

16 2.4 Thermodynamical aspects The energy achieved by solving the Schrödinger equation (Eq. 2) or by using density functional theory is called the electronic energy. This energy is found thanks to the Born-ppenheimer approximation, which states that the motion of the nuclei and that of the electrons of the atoms can be treated separately. Since the nuclei are much heavier than the electrons they can be assumed to be at a standstill relative to the smaller electrons and the electronic energy can thereby be calculated. In reality, however, the nuclei are never completely frozen, but always vibrate to some extent (even at 0 K). The lowest energy state of a molecule is therefore not the minima of the electronic energy surface. Instead it lies at a somewhat higher energy level, at the so called lowest vibrational state, as illustrated in Figure 5. This energy level can be evaluated by the use of a harmonic oscillator approximation (Eq. 4) and summation of all these extra energy contributions over the entire system gives the so called zero-point vibrational energy (ZPVE). This can be seen in Equation 5, where U is the internal energy and the rightmost term (the sum) is the ZPVE. E = (n + 1 )hω (4) 2 U = E electronic + i 1 2 hω i (5) Here n is the vibrational quantum number, ω is the vibrational frequency and h is the Planck constant. E Minimum of electronic E Lowest vibrational state Figure 5: A schematic potential energy curve showing the ground state electronic energy (in green) and its first vibrational state (in red). Additionally, the thermal effects for the system at temperatures above 0 K and can be found by taking the enthalpy H and the Gibbs free energy G into consideration. In order to obtain these results theoretically one needs to calculate all the vibrational frequencies of the system, which is a somewhat demanding calculation [8]. 16

17 2.5 Computational details Studied in this thesis was the last part of the reaction where HPPD converts HPP into HG (see Section 1) and the model chosen for the protein was based on the geometry optimization made by Wójcik et al. [1], during which only the atoms within 15 Å of the active-site iron were allowed to move. The calculations performed here were, thus, made on already defined structures and they were not changed during the project. In the paper [1] the main calculations were made on a system consisting of 6634 atoms, with 84 of them included in the QM part. However, another of the models tested, lacking one water molecule previously situated in the QM part, was found to give lower relative energies. Simulations showed that the water molecule seemed to move in to and out of the active site during the examination. In order to study how great an impact this mobility has on the energy barriers all calculations performed in this thesis were made on the system consisting of 6631 atoms. Figure 6 shows the QM model including said water molecule, in contrast to the inset of Figure 2 which was the QM model used for this thesis. Figure 6: The QM model including the water molecule (highlighted). The atomic coordinates were obtained from reference [1]. The initial calculations were performed with a lacvp double-ζ basis set on the mechanical embedding level, with AMBER as the force field method. Thereafter, the results were successively improved by the addition of different corrections or by the use of better ways to calculate the energy. This was done in order for us to get an idea of which 17

18 factors that affected the calculated energies to a greater extent than others. The elements that contributed the most to the final energies were investigated further with the aim of evaluating their sensitivity. The greater the effect, the more interesting the stability of their results are. Apart from the QM calculation on the part consisting of 81 atoms and the MM calculation on the rest of the enzyme, the factors investigated here were the following: Use of a larger basis set. The lacvp double-ζ basis set was replaced by a triple-ζ basis set combining the cc-pvtz(-f) basis for C, N, and H and the lacv3p+ basis and effective core potential for iron. Use of a dispersion correction. Two were tested, DFT-D2 and DFT-D3. Adding a thermal correction to the energy results. The one used here was based on a harmonic oscillator. Addition of a third solvation layer, in this case with the NIM-PCM. The potential energy curves presented in this report shows the energies of all structures in Figure 4 relative to that of the reactant (R). All calculations performed during this project were made with the NIM method of the Gaussian09 programme [25]. 18

19 3 Results The initial results were given by QM calculations performed with a double-ζ basis set on the mechanical embedding level, without any added corrections. Thereafter, the energies were successively improved by the addition of different correcting factors. The results of this study has been divided into two parts. The first one presents all of the investigated improvements and correction methods, and how great their effect to the final energy results were. Further, the second part deals with the sensitivity of the factors that were shown to affect the results more than the others. 3.1 Effects The effects investigated were the use of a larger basis set and addition of dispersion and thermal correction. Furthermore, the importance of using a QM/MM method instead of only calculating the QM energy, thus including the protein environment in the calculation, was analysed. The effect of adding a third solvation layer was also studied. An example of how the aforementioned effects affected the relative energy of one of the five structures, namely the second transition state (TS2 in Figure 4), can be seen in Figure 7. The initial QM calculation (on the part consisting of 81 atoms) of the energy of this structure was found to be 9.2 kcal/mol relative to the reactant and the final energy result was given by adding the found effects to the this energy. However, in order to take all observed effects into consideration the maximum effects obtained amongst all five structures, together with their absolute means (i.e. the average of the absolute values of the effects on all structures) are presented in Figure 8. 19

20 +7.6 E [kcal /mol] Basis set Dispersion Thermal MM PCM Figure 7: The separate effects of each studied correction method on the energy of the second transition state. Adding these to the initial QM calculations gives the final energy result E [kcal/mol] Basis set Dispersion Thermal MM PCM Figure 8: Maximum effect found when adding each correction method to the QM energy (in blue) and the absolute average values over all five structures, i.e. the mean of the absolute values of the effects found (in green). 20

21 The conclusions that can be drawn from studying the figures above are that the factor that seems to contribute the most to the complete reaction energy, apart from the initial QM calculation, is the energy given by the MM calculation (i.e. the effect of including the surrounding protein environment). Therefore we decided to investigate these two types of calculations further. This is presented in the following subsection. f the other studied factors it was the thermal correction that was found to influence the final result the most. The effect, however, was not considered large enough for any further enhancements of the method to be tested out, since the change in energy contribution could be assumed to become even smaller with the use of a more evolved correction scheme. Moreover, the accompanying frequency calculation already resulted in a considerable increase in required computation time. In conclusion, using a thermal correction based on a harmonic oscillator seem to be enough when treating systems like the one investigated here. Adding a dispersion correction was on the other hand quite a simple matter. For this reason, although the energy effect of the addition was found to be even smaller than for the thermal correction, two dispersion correction methods were tested: DFT-D2 and DFT-D3. The effects did not give particularly different energy results (the difference was around 1 kcal/mol), which is why it was concluded that any of the two could be used and still give reasonable results. The use of a larger basis set did not affect the energies much and the required computation times where increased considerably. Increasing the basis set even further could be expected to affect the result even less, which is why it was deemed unnecessary. However, the effect was still large enough for us to conclude that the calculations should be performed with the use of a triple-ζ, and not just a double-ζ, basis set in order to give results that are plausible. Finally the addition of a solvation layer did barely affect the calculated energies at all, and just as for the thermal correction and the larger basis set it resulted in a significantly longer calculation time. Consequently, this effect was found to be insignificant and was not investigated further. 3.2 Sensitivity The results presented in the previous subsection suggested two areas that contributed to the final energies significantly and, hence, would be interesting to study further: the calculated QM energy and the effect of including the surrounding protein into the calculation. These were investigated by varying the choice of functional and by extending the QM part to include a greater part of the protein. More details follow below. 21

22 3.2.1 QM energy The sensitivity of the QM energy was explored by studying how the relative energies were affected by different types of DFT functionals. In Figure 9, the energies of the two transition states relative to R (Fig. 4), given by each functional studied, have been plotted against the amount of HF exchange included in that particular method. The functionals studied were all that were listed in table 2 (Section 2) and the B3LYP functional with different amounts of HF exchange, ranging from 0 to 25 % in increments of 5 %. 30 BLYP TPSS BP86 M06-L TS1 B3LYP TS1 ther TS2 B3LYP TS2 ther E [kcal/mol] BLYP TPSS BP86 M06-L TPSSh TPSSh M06 M HF exchange Figure 9: Dependence of the relative energies of the transition states on the amount of exact exchange in the functional. The calculations were made with double-ζ basis set at the mechanical embedding level with each of the functionals listed in Table 2 (Section 2). B3LYP-D2 dispersion and thermal corrections were added to all functionals. Figure 9 shows that the relative energies of the transition states varied considerably with different choices of functional used for the calculations. Values of the first transition state were found to differ around 10 kcal/mol, whereas the effect on the second transition state was nearly three times as high. However, since there are functionals that are more suited for these kinds of systems than others, it might be interesting to study the effect on the energy barriers by comparing only the functionals that one might actually choose. When comparing only the functionals TPSSh, M06 and B3LYP with 10, 15 and 20 % exact exchange (that is, B3LYP ±, B3LYP* and B3LYP) the energy spans became quite a lot smaller (around 9 and 13 kcal/mol for the first and the second transition state, respectively), although the effect was still substantial. 22

23 3.2.2 Protein environment The effect of including the protein environment through MM calculations shows how great effect the surrounding protein has on the active site (approximated by the QM part). Figure 8 shows that this effect was large in this particular system (around 8 kcal/mol). For this reason it was concluded that extending the QM part could improve the results, since including a larger part of the protein into the more accurate QM calculation should describe reality in a better way. The parts to include in the expanded QM model were found by analysing how much each amino acid moved during the reaction (thus indicating their importance to the process). Figure 10 shows part of the protein coloured according to how much each atom moved when going from R to TS1. Red indicates very little movement (less than 0.1 Å), grey somewhat more and blue means that the atoms have moved a lot. The atom that moved the most for this particular structure had moved around 2.7 Å. The atoms represented with a ball-and-stick model are the ones included in the small QM model and the thicker licorice shaped parts represent the atoms added to the extended QM model. In addition, the thin licorice shaped parts show other atoms that also moved to some extent during the reaction, but that was not included in any QM calculation. Figure 10: Part of the protein coloured according to movement between R and TS1 (blue means a lot and red little). Representation of small QM part in ball-and-stick model and extended QM part in thicker licorice shape. The rest are parts that moved somewhat without being included in any QM part. 23

24 Figure 11 shows how the relative energies of the mechanism were affected when applying the larger QM model and in this case all energy values were decreased. E [kcal/mol] R TS1 INT1 TS2 P Reaction coordinate Figure 11: Potential energy curves for the studied mechanism found with two different QM models: smaller in blue and larger in red. The calculations were made with a triple-ζ basis set at the EE level with D2 dispersion and thermal correction. The new QM model consisted of 184 atoms instead of 81 and, as expected, adding this extra part of the protein to the more accurate QM calculation was found to reduce the MM contribution to the total energy considerably (to near 1 kcal/mol). In order to really understand what it was that changed when the greater QM part was applied, the separate contributions to the complete energy were investigated. Figures 12 and 13 each show these contributions, i.e. the MM energy (divided into Coulomb interactions, van der Waals interactions and bonded terms), QM energy (corrected for dispersion) and the thermal correction, for both QM models. However, these calculations were performed before the effect of the larger basis set was studied, which is why the energy values in Figure 11 (performed with a triple-ζ basis set) and those of Figs. 12 and 13 (double-ζ) do not match. In general one can see that for the small QM model the greatest contribution to the MM energy comes from the van der Waals interactions between the two subsystems and that, apart from the second transition state, the Coulomb interactions for each structure are negligible. n the other hand, in the large QM model the latter are of greater importance, while the significance of the former is somewhat diminished. ne can also conclude that the small MM contribution in the large model is due to significant cancellation of the two aforementioned interactions. 24

25 20 E [kcal/mol] R TS1 INT1 TS2 P Coulomb vdw Bonded QM Thermal Figure 12: The potential energy curve for the small QM model with its respective contributions. The calculations were performed with a double-ζ basis set and a D2 dispersion correction was added to the QM energy bar. 20 E [kcal/mol] R TS1 INT1 TS2 P Coulomb vdw Bonded QM Thermal Figure 13: The potential energy curve for the large QM model with its respective contributions. The calculations were performed with a double-ζ basis set and a D2 dispersion correction was added to the QM energy bar. 25

26 Since the Coulomb energy contribution increased significantly with the use of a larger QM model, the electrostatic interactions were studied closer by applying an electronic embedding scheme to both models. The results showed that the energies found with the small QM model were not changed notably. However, the reaction energies of the large QM model were all increased. Figure 14 shows how the relative energies found in Figure 13 are affected by the change of embedding scheme. For example, the most affected structures were found to be TS2 and P, with an increase of around 4-5 kcal/mol. This seems rather reasonable since the change in Coulomb interaction energy when switching model was larger for TS2 and P than for the other two structures. E [kcal/mol] 20 R TS1 INT1 TS2 P Reaction coordinate Figure 14: The potential energy curves of the studied reaction mechanism for the large QM model, calculated with two different embedding schemes: mechanical in purple and electronic in orange. The calculations were performed with a double-ζ basis set with an added D2 dispersion and thermal correction. 3.3 Model Another effect studied was the choice of model. The results in the recently published paper [1] suggested that one of the water molecules included in their QM part should not be there due to its mobility in the active site (see Figure 6). In order to evaluate the effect that said water molecule had on the results all calculations in this thesis were performed on a model lacking it. That way comparisons between the two models were made possible. The effect that the difference had on the relative energies can be seen in Figure 15 and the observations that can be made are that excluding the water molecule lowered all calculated reaction energies. The product, in particular, (and by extent also the second transition state) seems a lot more stabilized relative to the reactant. 26

27 E [kcal/mol] Reaction coordinate Figure 15: The potential curve for the studied reaction mechanism for the model with (in blue) and without water (in purple). The calculations were performed with a triple-ζ basis set on the small QM model, with added D2 dispersion and thermal correction. 27

28 4 Discussion ne of the questions of this study was whether the conclusions regarding the reaction mechanism of HPPD drawn in the recently published paper [1] were reasonable or if any of the other proposed reaction mechanisms could be equally probable. As presented in the Introduction the energy difference between the first transition state of the suggested mechanism (A) and that of mechanism B was found to be just below 9 kcal/mol. Results suggesting that the energies found (especially those for TS1) could be increased by half of this amount at the same time as the energies of mechanism B is equally decreased would therefore indicate that further investigations are required before any actual conclusions can be drawn. Another aspect that was going to be investigated was whether it would be possible to say something about which of the studied reaction barriers (i.e. the two transition states) is the highest one, thus representing the rate limiting reaction step. The first of the following subsections contains comments on what is important to think about in theoretical studies. Moreover, the three factors that were found to affect the results to a significant extent are also discussed below. Finally, the section is ended with some final remarks and a short summary of the conclusions drawn here. 4.1 Recommendations The results presented in the previous section tell us what effects one should think about in order to get as trustworthy results as possible. Following is a list of the ones that have been considered in this thesis, in order of importance: 1. Functional choice 2. Description of protein environment 3. Selection of model 4. Thermal correction 5. Dispersion correction 6. Basis set 7. Solvation model Although only one enzyme was investigated here, these results may be used as a guideline when studying similar systems. The three first factors will be discussed more thoroughly below, but it is obvious that they need to be considered in order to get satisfying results. Number 4 to 6 affected the results to some extent, but was decided to be stable enough to describe the system in a good way (i.e. one should use a thermal approximation based on the harmonic oscillator, one of the two Grimme dispersion corrections and a triple-ζ 28

29 basis set for these types of calculations). The use of a solvation model did not affect the results notably and did also increase the computation time considerably, which is why our conclusion is that it is not a very important effect to consider for these kinds of systems. 4.2 Choice of functional Although it is difficult to say which is the most appropriate of all available functionals we can draw conclusions concerning the results by comparing their different performances. As presented in Figure 9 the energy of the first transition state was found to vary with about 10 kcal/mol depending on the functional chosen. However, the functional used in the paper (B3LYP) [1] was found to give values in the middle of the observed energy span. Thus, by changing the functional the energy of the first transition state may vary within ±5 kcal/mol. Unless the functional causing the energy value of TS1 to increase by this amount affect the transition state of mechanism B the opposite way this is within the limits of what is acceptable before the another mechanism becomes of interest. In contrast, it is not possible to conclude what the rate limiting step is. As a matter of fact, the relative energy of the second transition state was the higher than the first for all functionals considered here except for the one used in the paper (B3LYP) and one other functional. Since we do not know with certainty which the correct functional is, we cannot say whether the first or the second barrier is the highest. The results show that the sensitivity to the choice of functional may vary greatly depending on the structure studied (as our two transition states did). Since this is something that cannot be known beforehand our recommendation would be to always test the sensitivity of the system by performing calculations with different functionals (preferably more than two). This way, while not knowing which the most appropriate functional is, one may still know if choosing different ones might affect the results significantly. n the other hand, in order to determine what the rate limiting step is more thorough investigations might be needed, since different structures within the same reaction mechanism can be affected differently by the same functional. 4.3 Size of QM part The extension of the QM part from 81 to 184 atoms lowered the reaction energies of the structures with almost 6 kcal/mol at most (as a matter of fact, the maximum change was that of the first transition state). A decrease of 6 kcal/mol would mean that the transition state of mechanism B would have to be lowered with at least 15 kcal/mol to be equally possible, which is not very probable. The rate limiting step was not changed by extending the QM part (Figure 11), i.e. the calculations performed with the B3LYP functional still gave results where the first transition state was the highest. In conclusion, what can be said is that although the change of QM model affected the results, it did not make any difference in terms of the conclusions drawn in the paper. 29