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3. Simulation of enzyme catalyzed reactions
The thermodynamic, kinetic and dynamic features of a chemical process are determined by its free energy surface, i. e. the energy of a system as a function of the nuclear coordinates. The knowledge of the complete free energy surface of a reaction in an enzyme environment would reveal the reaction path followed by the enzyme when converting substrates to products. The comparison of the free energy surfaces of the enzyme reaction and a corresponding reference reaction in aqueous solution could show, by which means the enzyme accelerates a chemical reaction. It is however not possible to establish a full free energy surface of an enzyme catalyzed reaction by experimental techniques. This task could, at least in principle, be fulfilled by computer simulation studies.
In this chapter I will present the basic concepts of enzyme catalysis and kinetics. Then the principles and main problems of computer simulations of enzymatic reactions are discussed. In the last section the widely used empirical valence bond theory of Arieh Warshel is described in detail. An application of this theory on the enzyme Acetylcholinesterase is presented in chapter 4.
3.1. Theory of enzyme catalysis
Catalysis is in general defined as a process where the rate of a chemical reaction is increased by the interaction of the reacting partners with another substance which is called the catalyst. The equilibrium constant of the reaction is not changed by such a process. The catalyzed reaction shows a free energy surface with an activation barrier on the reaction path that is lower than in the uncatalyzed reaction. The modification of the free energy surface results from the interaction with the catalyst.
The catalytic efficiency of enzymes is outstanding. The rates of enzyme catalyzed reactions can be enhanced by a factor of up to 1017 compared to the uncatalyzed reaction.^33 The proposition by Emil Fischer that an enzyme and its substrate can be seen in analogy to a lock and a key was the first attempt to account for the exceptional power and specificity of enzymes as biological catalysts.
The rate enhancement of an enzyme catalyzed reaction is today explained by the fact, that the transition state of the reaction (rather than the substrate) fits very well to the enzyme active site.^34 Though the term fit goes beyond a simple geometric description. It is widely accepted that enzymes work by stabilizing the transition states of chemical reactions, but it is not clear in full detail by which means this task is fulfilled.
Chemists use in general acids, bases and nucleophiles as catalysts. Enzymes use the same chemistry but are more efficient due to a very subtle and equilibrated binding of substrates, transition states and products. The catalytic effect of enzymes is assumed to consist of several contributions but their relative significance is still discussed. In the next section I will introduce the basic parameters of enzyme kinetics and thermodynamics and show how they can be used to explain the principles of enzyme catalysis.
- Simulation of enzyme catalyzed reactions
3.1.1. Basic enzyme kinetics
In contrast to an uncatalyzed chemical reaction in solution an enzyme catalyzed reaction occurs after the substrate has bound to the enzyme. Therefore, the binding of the substrate, the actual chemical conversion and the release of products determine the overall reaction rate. The reaction scheme of a typical enzymatic reaction is given in Fig. 3.1:
E + S ES P + E
k 1
k -
kcat EP
k 2
k -
Figure 3.1.: Schematic representation of a typical enzymatic reaction. Key: E=enzyme, S=substrate, ES=enzyme-substrate complex, EP=enzyme-product complex, P=product. Each process has a reaction rate constant ki , where k 1 and k − 1 determine the binding of the substrate and k 2 and k − 2 determine the release of the product. The rate constant kcat determines the chemical conversion from substrate to product.
At low substrate concentration the velocity of the reaction is given by
v = kcat KM
[ ET ][ S ] (3.1)
where [ ET ] is the total enzyme concentration and KM , the Michaelis constant, is defined as
KM =
k − 1 + kcat k 1
If kcat is very small compared to k − 1 , the Michaelis constant can be approximated by the dissoci- ation constant KD of the enzyme substrate complex:
KM ≈ KD =
[ E ][ S ]
[ ES ]
From Eq. 3.1 the ratio kcat / KM can be considered as a second order rate constant. It is determined by both processes responsible for the overall reaction rate, which are substrate binding and the chemical conversion. This holds if the release of products is fast enough such that it does not block the binding of new substrate. The smaller KM , the better the substrate binds to the enzyme. The chemical process of transforming the substrate to product is characterized by the nominator kcat. The transformation does not need to be a one step reaction, but can consist of a sequence of elementary processes, that all contribute to kcat. The term kcat / KM is the most critical parameter for enzyme kinetics. As the enzyme and the substrate cannot encounter more rapidly than diffusion in their solution permits, the velocity of the enzymatic reaction has an upper limit, which is ∼ 109 s −^1 M −^1. The significance of kcat / KM becomes obvious from the following interpretation of Fig. 3.2: The overall barrier of an enzymatic reaction corresponds to the energy difference between E + S and ES ‡. This energy difference is determined by the values of ∆ Gbind and ∆ Gcat. The binding energy ∆ Gbind is related to the Michaelis constant by the equation
∆ Gbind = RT lnKM. (3.4)
- Simulation of enzyme catalyzed reactions
An enzyme-substrate pair has a small KM value, if the enzyme binds the substrate well. Moreover, the better the stabilization of the actual transition state in the enzyme, the smaller is the activation free energy ∆ G ‡ cat. This activation energy is related to the rate kcat of the reaction via an expression from transition-state theory.:^35
kcat = kBT h
e −∆ G ‡β (3.5)
where h is Planck´s constant. The overall barrier of the enzymatic reaction is therefore small if kcat / KM is large. Enzymes, that need to perform their catalytic function predominantly fast, have evolved to enlarge the ratio kcat / KM. As enzymes have to bind the transition state of a reaction as good as possible, the binding of the structurally different substrate cannot be optimal. Accordingly an enzyme should not evolve to yield a perfectly small KM for its substrate. Preferable it should bind the substrate relatively weakly. One part of the reduction of the overall barrier in the enzyme compared to aqueous solution can be accomplished via the substrate binding energy ∆ Gbind. This would stabilize ES and ES ‡ by the same amount. However, the binding energy alone cannot account for the overall reduction of ∆ G ‡^ as KM should not be too small. This means, that ∆ G ‡ cat has to be reduced considerably compared to ∆ G ‡ cage in aqueous solution. The most important part of the catalytic effect results
from this reduction of ∆ G ‡ cat by transition state stabilization. In general it is not clear what is the nature and mechanism of this stabilization. Enzyme transition states cannot be isolated experimentally and they cannot be understood completely without some quantitative model for structure-function correlation.
3.1.2. The chemistry of enzyme catalyzed reactions
A chemical reaction can be catalyzed in aqueous solution by a base or an acid. In Fig. 3.3 an example for general base catalysis is given.
R'
O OR
H O^ H
R' O OR
H O^ H
δ -
δ+ TS
(a)
R' O OR H O^ H O CH 3 O
R'^ O OR H O O CH 3 O
δ -
δ (^) -
TS
H
general base
(b)
Figure 3.3.: The nucleophilic attack of a water molecule on the carbonyl C-atom of an ester. Figures (a) shows the uncatalyzed reaction. (b) The reaction is catalyzed by a general base that stabilizes the transition state(TS) by partly abstracting a proton from the water molecule. The negative charge is delocalized. The abstraction of the proton increases also the nucleophilicity of the water molecule.
3.1. Theory of enzyme catalysis
In Fig. 3.3 (a) the uncatalyzed hydrolysis of an ester is shown. The transition state shows a positive and a negative partial charge. The transition state can be stabilized by the transfer of a proton from the water molecule to a general-base as shown in Fig. 3.3 (b). In analogy general-acid catalysis can be applied to stabilize a negative charge by a proton transfer from an acid. The term general-base or general-acid catalysis is used to distinguish it from specific-base or specific-acid catalysis, where the catalyst is the hydroxide-ion or the proton. Acid and base catalysis is applied by chemists in solution as well as by enzymes. Examples of general-acid or base catalyzed reactions can be found in many enzymes e. g. in triose phosphate isomerase,36, 37^ where a histidine works as general-acid and a glutamate as general-base. Histidine works also as general-base in serine proteases^38 like trypsin, chymotrypsin and subtilisin and in cysteine proteases^39 like papain and actinidin. In the above example of general-base catalysis, the acceleration of the reaction is accomplished without changing the mechanism. Another very efficient means to catalyze a reaction is the modification of the reaction pathway. Examples for this type is the nucleophilic catalysis in acyl transfer- or hydrolytic reactions. In Fig. 3.4 the hydrolysis of acetic anhydride is shown. The rate of reaction is enhanced by pyridine, because the highly reactive acetylpyridinium-ion is formed rapidly. Nucleophilic catalysis can only be efficient, if the nucleophile that acts as catalyst (in Fig 3.4 pyridine) is more nucleophilic than the one it replaces (in our example acetate). Moreover the intermediate must be more reactive than the parent compound.
H 3 C O^ CH 3
O O
N
H 3 C
O N+ NH+
CH 3 COO-
H 2 O CH 3 COOH
pyridine acetylpyridinium ion
acetic anhydride
Figure 3.4.: The hydrolysis of acetic anhydride catalyzed by pyridine. This reaction is an example of catalysis, where the mechanism of the reaction is changed an a reactive intermediate is formed. As pyridine attacks the carbonyl carbon as nucleophile, it performs nucleophilic catalysis
Another contribution to the catalytic properties of enzymes might come from entropic effects. Uncatalyzed reactions may be unfavorable due to the loss of entropy, when the reacting partners form an adduct that can undergo the corresponding reaction. It was suggested that an enzyme uses the energy of substrate binding to compensate for the loss of entropy.40, 41^ This theory is supported42, 43^ but also questioned.44, 45^ It is difficult to quantify the entropy in enzyme catalyzed reactions by experiments and also by theoretical calculations and simulations. A qualitative exam- ination that investigated the entropic contributions to the catalysis in serine proteases indicated that these contributions are rather small.^45
3.1.3. Electrostatic contributions in enzyme catalysis
It is widely accepted that electrostatic forces are perhaps the most relevant factors in enzyme catalysis. As many classes of enzymatic reactions involve large changes in the charge distribution of the reacting species it is likely that enzymes evolved to stabilize the charge distribution of the corresponding transition states. This indicates the importance of electrostatic interactions. The stabilization of charge distributions should be accomplished by enzymes better than by water,
3.1. Theory of enzyme catalysis
Carbonic Anhydrase
Carbonic anhydrase catalyzes the reversible hydration of carbon dioxide and the dehydration of hydrogen carbonate. The mechanism involves two processes: The activation of a zinc-bound water molecule, that donates a proton to His64 and a nucleophilic attack of the resulting hydroxide ion on carbon dioxide yielding hydrogen carbonate. The active site of carbonic anhydrase contains a zinc ion. (see Fig. 3.6) Its crucial role was determined quantitatively by showing that the cation reduces the pK a -value of the bound water molecule by providing electrostatic stabilization of the hydroxide ion. The pK a value of the bound water molecule in carbonic anhydrase is between 7 and 8.^51 Additionally the active site of the enzyme stabilizes the HCO− 3 -ion by electrostatic interactions.^52
HN N
Zn H N N NH
N
O O O
O N O^ O
N O
N^ H N His
OH 2
His His
Glu
Glu
Thr
Gln
Figure 3.6.: A schematic representation of the active site of carbonic anhydrase. The zinc bound water molecule is activated and transfers a proton to His64. Then the result- ing hydroxide ion attacks a carbon dioxide molecule yielding hydrogen carbonate. The active site stabilizes the hydroxide ion as well as the hydrogen carbonate by electrostatic interactions.
3.1.4. Reference reactions
The efficiency of enzyme catalysis can only be evaluated by comparing the catalyzed reaction with a suitable reference reaction. In section 3.1.1 we saw that the reduction of ∆ G ‡ cat is very important for enzyme catalysis. See also Fig. 3.2 that illustrates this effect. A recent analysis of the reaction of ribonuclease indicated for example that this enzyme provides a transition state stabilization ∆ G ‡ cage − ∆ G ‡ cat of 18 kcal/mol.^53 In my work I concentrate on the investigation of this kind of stabilization i. e. the difference between the activation barrier of a reaction in a solvent cage and the reaction in the enzyme active site. The reaction in the solvent cage is a very good reference state because
- one can choose a reaction that has the same mechanism as the reaction in the enzyme. A possible difference in mechanism between the uncatalyzed reaction in water and the enzyme catalyzed reaction is known as a catalytic effect, but its investigation is not the aim
- Simulation of enzyme catalyzed reactions
of my work as this effect is well understood in terms of nucleophilicity or pK a -values (see section 3.1.2).
- possible concentration effets of reactions in aqueous solution cancel. The concentrations of the reacting species are the same in a solvent cage and in the enzyme active site.
- the focus is then concentrated on the actual task of enzymes: the stabilization of transition states and the advantage of the enzyme compared to water in this respect. In this work I apply computer simulation studies to compare chemical reactions in aqueous solution and in enzymes. The theoretical tools for the simulation of chemical reactions in enzymes by computers are far from being usable as black box programs. The field is under intensive development and no standard algorithms or software have evolved. In the next section I will outline the general principles applied in these simulation studies.
3.2. How to simulate an enzymatic reaction?
To obtain the free energy surface of a reacting system it is necessary to sample all relevant conformations of the system. Such a search in the configurational space can be done with Monte- Carlo (MC) or Molecular Dynamics (MD) simulations. These methods generate conformations corresponding to energy potentials e. g. a mechanical force field.54, 55, 56 Enzymes are very large molecules and MD or MC studies on them are very time consuming even when simple potential functions are used as force field. To sample the conformational space of an enzyme substrate complex sufficiently to draw reliable conclusions, it is necessary to use a thoroughly parameterized classical molecular mechanics (MM) force field as it is described in appendix C. This involves always the evaluation of a large number of interactions between all atoms at each conformation. With fast computers and efficiently implemented algorithms it is nevertheless possible today to simulate an average protein in solution up to several ns.^57 Even simulations of up to 1 ms have been performed, but this is still an exception.^58 Another situation arises with the simulation of chemical reactions. Their study requires quan- tum mechanical (QM) methods, as only quantum mechanics is able to describe bond breaking and bond formation processes. The computational time that is needed for QM calculations increases with the number of explicitly treated electrons N proportional to N^3 or N^4. From the viewpoint of computational costs it is impossible until today and also for the near future to apply full QM calculations on a whole enzyme-substrate complex involving at least some thousand atoms. A way out of this problem is the combination of quantum mechanical (QM) and molecular mechanical (MM) methods. This concept was first introduced in 1976 by Warshel and Levitt.^59 It devides the system into two regions: The reacting fragment is described by quantum chemical methods, whereas the environment is described classically. Since the initial idea of the QM/MM approach has been presented, various models of this principle have been published and were applied to enzymatic reactions.60, 61, 62, 63^ The QM/MM method has recently been reviewed by Field.^64 The individual models differ in the special QM and MM methods used as well as in the treatment of the QM/MM interactions. The general Hamiltonian for a system consisting of differently described parts can be formulated as
H = HQM + HMM + HQM / MM (3.6)
where HQM represents the Hamiltonian for the quantum system, HMM the interaction within the surrounding medium and HQM / MM the interaction between the quantum region and the surround- ing medium. The level of quantum chemical methods employed range from rather accurate ab
- Simulation of enzyme catalyzed reactions
Conventional force fields describe molecular systems with an energy function corresponding to one chemical state only. They do not provide any relation between the energies of resonance states possessing different bonding schemes, that result in alteration of Lennard-Jones or charge parameters. The energy difference between two resonance states is given by the difference of heat of formation in the gas phase and is thus introduced in the energy function by the constant energy parameter α( μ ). The described energy function is used to sample the configurational space of the system by molecular dynamics. Subsequently, the adiabatic electronic ground state energy E g^1 ,^2 , mediating the transition between the states 1 and 2, is calculated at all configurations:
E g^1 ,^2 = E g^1 ,^2 (ε 1 − ε 2 ) =
(ε 1 + ε 2 ) −
(ε 1 − ε 2 )^2 + 4 H 122 (3.9)
Eg is the lower of the two eigenvalues of a 2 × 2 Hamiltonian matrix, where the diagonal elements are the potentials ε 1 and ε 2 , from Eq. 3.8 and the off-diagonal element H 12 is represented by H 12 = A 12 exp [ μ 12 ( rXY − r ‡ XY ) − η 12 ( rXY − r ‡ XY )^2 ] (3.10)
where rXY is the distance between two atoms that are involved in a bond breaking or forming process in the reaction considered and can be used to monitor to which extent the reaction has evolved. The distance r ‡ XY is a constant, chosen such that H 12 adopts a maximal value close to the transition state. The free energy difference of the transition of state 1 and state 2 is evaluated with the free energy perturbation approach (FEP):^67
∆ Gmap (λ (^) j ) = Gmap (λ (^) j ) − Gmap (λ 0 ) = − kBT
j − 1 ∑ i = 0
ln 〈 exp [−(ε map (λ i + 1 ) − ε map (λ i ))/ kBT ]〉 i (3.11)
∆ Gmap (λ (^) j ) represents the free energy associated with moving on the mapping potential ε^1 map ,^2 (λ) defined in Eq. 3.7. The generalized reaction coordinate X, is defined as the energy gap ε 1 − ε 2 of the two energy functions at each conformation of the trajectory. The reaction coordinate X is partitioned in a number of bins Xm , typically 50. The free energy, ∆ G^1 ,^2 ( X ), corresponding to the trajectories moving on the adiabatic energy surface of the electronic ground state Eg (Eq. 3.9) mediating the transition from the reactant state to the product state, is obtained with the umbrella sampling expression
exp [−∆ G^1 ,^2 ( Xm )/ kBT ] = 1 N ( Xm ) ∑ j n ( j , Xm ) exp [−∆ G^1 map ,^2 (λ (^) j )/ kBT ] · 〈 exp [−( E g^1 ,^2 ( Xm ) − ε map (λ (^) j ))/ kBT ]〉 (^) j , Xm
where N ( Xm ) = (^) ∑ j
n ( j , Xm ) (3.12)
The Boltzmann factor of the free energy ∆ G^1 ,^2 ( Xm ) is obtained as a weighted sum over the different λ (^) j ensembles, n ( j , Xm ) is the number of conformations from the λ (^) j ensemble, where the reaction coordinate X belongs to bin m. The ensemble average in Eq. 3.12 considers all conformations, where the reaction coordinate corresponds to bin m and λ (^) j. This statistical average accounts for the energy difference between the mapping potential ε^1 map ,^2 used for the MD simulation and the adiabatic potential energy surface E^1 g ,^2 of the reaction considered.
3.3. The empirical valence bond method
An important aspect of the EVB procedure is, as stated above, the calibration of the EVB parameters in H 12 and of α. The parameters are chosen such that in a simulated reference reaction in aqueous solution the resulting free energy profile reproduces the experimental value of the activation barrier and of the free reaction energy. In practice, the reaction of interest is simulated in a water sphere and the obtained energy data are evaluated within the FEP procedure, in which the parameters α and H 12 can be varied to yield the proper free energy curve. A detailed discussion about setting up the reference energy profile is given in section 4.3.2, within the discussion of the simulation of the deacylation step in acetylcholinesterase. These calibrated parameters are then also used to evaluate the energy profiles from a simulation in which the water sphere has been replaced by the enzyme and water molecules to fill the protein cavities. With this approach, the effect of the enzyme environment on the free energy reaction profile is obtained.
