Glutathione transferase A1-1: catalytic role of water

For more than almost 30 years now, glutathione transferases (GSTs) have been known as xenobiotic/endobiotic detoxification enzymes. GSTs catalyze the nucleophilic addition of glutathione (GSH) sulphur thiolate to a wide range of electrophilic substrates, building up a less toxic and more soluble compound, which can then be removed from the cell. Recently we proposed a consistent GSH activation mechanism. By performing QM/MM calculations, we demonstrated that a water molecule, following a first conformational rearrangement of GSH, is capable of assisting a proton transfer between the GSH thiol and alpha carboxylic groups. In this study we go further in the analysis of the water role in GSH activation by performing a long Molecular Dynamics (MD) study on glutathione transferase A1-1 Thr68 mutants complexed with GSH and the GSH decarboxylated analogue (dGSH), for which experimental kinetic data are available.     

A critical evaluation of different QM/MM frontier treatments with SCC-DFTB as the QM method

The performance of different link atom based frontier treatments in QM/MM simulations was evaluated critically with SCC-DFTB as the QM method. In addition to the analysis of gas-phase molecules as in previous studies, an important element of the present work is that chemical reactions in realistic enzyme systems were also examined. The schemes tested include all options available in the program CHARMM for SCC-DFTB/MM simulation, which treat electrostatic interactions due to the MM atoms close to the QM/MM boundary in different ways. In addition, a new approach, the divided frontier charge (DIV), has been implemented in which the partial charge associated with the frontier MM atom ("link host") is evenly distributed to the other MM atoms in the same group. The performance of these schemes was evaluated based on properties including proton affinities, deprotonation energies, dipole moments, and energetics of proton transfer reactions. Similar to previous work, it was found that calculated proton affinities and deprotonation energies of alcohols, carbonic acids, amino acids, and model DNA bases are very sensitive to the link atom scheme; the commonly used single link atom approach often gives error on the order of 15 to 20 kcal/mol. Other schemes give better and, on average, mutually comparable results. For proton transfer reactions, encouragingly, both activation barriers and reaction energies are fairly insensitive (within a typical range of 2-4 kcal/mol) to the link atom scheme due to error cancellation, and this was observed for both gas-phase and enzyme systems. Therefore, the effect of using different link atom schemes in QM/MM simulations is rather small for chemical reactions that conserve the total charge. Although the current study used an approximate DFT method as the QM level, the observed trends are expected to be applicable to QM/MM methods with use of other QM approaches. This observation does not mean to encourage QM/MM simulations without careful benchmark in the study of specific systems, rather it emphasizes that other technical details, such as the treatment of long-range electrostatics, tend to play a more important role and need to be handled carefully.     

Catalytic roles of the metal ion in the substrate-binding site of coenzyme B12-dependent diol dehydratase

Functions of the metal ion in the substrate-binding site of diol dehydratase are studied on the basis of quantum mechanical/molecular mechanical (QM/MM) calculations. The metal ion directly coordinates to substrate and is essential for structural retention and substrate binding. The metal ion has been originally assigned to the K(+) ion; however, QM/MM computations indicate that Ca(2+) ion is more reasonable as the metal ion because calculated Ca-O distances better fit to the coordination distances in X-ray crystal structures rather than calculated K-O distances. The activation energy for the OH group migration, which is essential in the conversion of diols to corresponding aldehydes, is sensitive to the identity of the metal ion. For example, the spectator OH group of substrate is fully deprotonated by Glu170 in the transition state for the OH group migration in the Ca-contained QM/MM model, and therefore the barrier height is significantly decreased in the model having Ca(2+) ion. On the other hand, the deprotonation of the spectator OH group cannot effectively be triggered by the K(+) ion. Moreover, in the hydrogen recombination, the most energy-demanding step is more favorable in the Ca-contained model. The proposal that the Ca(2+) ion should be involved in the substrate-binding site is consistent with an observed large deuterium kinetic isotope effect of 10, which indicates that C-H bond activation is involved in the rate-determining step. Asp335 is found to have a strong anticatalytic effect on the OH group migration despite its important role in substrate binding. The synergistic interplay of the O-C bond cleavage by Ca(2+) ion and the deprotonation of the spectator OH group by Glu170 is required to overcome the anticatalytic effect of Asp335.     

The fragmentation-recombination mechanism of the enzyme glutamate mutase studied by QM/MM simulations

The radical mechanism of the conversion of glutamate to methylaspartate catalyzed by glutamate mutase is studied with quantum mechanical/molecular mechanical (QM/MM) simulations based on density functional theory (DFT/MM). The hydrogen transfer between the substrate and the cofactor is found to be rate limiting with a barrier of 101.1 kJ mol(-1). A careful comparison to the uncatalyzed reaction in water is performed. The protein influences the reaction predominantly electrostatically and to a lesser degree sterically. Our calculations shed light on the atomistic details of the reaction mechanism. The well-known arginine claw and Glu 171 ( Clostridium cochlearium notation) are found to have the strongest influence on the reaction. However, a catalytic role of Glu 214, Lys 322, Gln 147, Glu 330, Lys 326, and Met 294 is found as well. The arginine claw keeps the intermediates in place and is probably responsible for the enantioselectivity. Glu 171 temporarily accepts a proton from the glutamyl radical intermediate and donates it back at the end of the reaction. We relate our results to experimental data when available. Our simulations lead to further understanding of how glutamate mutase catalyzes the carbon skeleton rearrangement of glutamate.     

Mechanism of proteolysis in matrix metalloproteinase‐2 revealed by QM/MM modeling

The mechanism of enzymatic peptide hydrolysis in matrix metalloproteinase‐2 (MMP‐2) was studied at atomic resolution through quantum mechanics/molecular mechanics (QM/MM) simulations. An all‐atom three‐dimensional molecular model was constructed on the basis of a crystal structure from the Protein Data Bank (ID: 1QIB), and the oligopeptide Ace‐Gln‐Gly～Ile‐Ala‐Gly‐Nme was considered as the substrate. Two QM/MM software packages and several computational protocols were employed to calculate QM/MM energy profiles for a four‐step mechanism involving an initial nucleophilic attack followed by hydrogen bond rearrangement, proton transfer, and C? N bond cleavage. These QM/MM calculations consistently yield rather low overall barriers for the chemical steps, in the range of 5–10 kcal/mol, for diverse QM treatments (PBE0, B3LYP, and BB1K density functionals as well as local coupled cluster treatments) and two MM force fields (CHARMM and AMBER). It, thus, seems likely that product release is the rate‐limiting step in MMP‐2 catalysis. This is supported by an exploration of various release channels through QM/MM reaction path calculations and steered molecular dynamics simulations. © 2015 Wiley Periodicals, Inc.     

Peptide hydrolysis catalyzed by matrix metalloproteinase 2: a computational study

The MMP-2 reaction mechanism is investigated by using different computational methodologies. First, quantum mechanical (QM) calculations are carried out on a cluster model of the active site bound to an Ace-Gly approximately Ile-Nme peptide. Along the QM reaction path, a Zn-bound water molecule attacks the Gly carbonyl group to give a tetrahedral intermediate. The breaking of the C-N bond is completed thanks to the Glu 404 residue that shuttles a proton from the water molecule to Ile-N atom. The gas-phase QM energy barrier is quite low ( approximately 14 kcal/mol), thus suggesting that the essential catalytic machinery is included in the cluster model. A similar reaction path occurs in the MMP-2 catalytic domain bound to an octapeptide substrate according to hybrid QM and molecular mechanical (QM/MM) geometry optimizations. However, the rupture of the Gly( P 1) approximately Ile( P 1') amide bond is destabilized in the static QM/MM calculations, owing to the positioning of the Ile( P 1') side chain inside the MMP-2 S 1' pocket and to the inability of simple energy miminization methodologies to properly relax complex systems. Molecular dynamics simulations show that these steric limitations are overcome easily through structural fluctuations. The energetic effect of structural fluctuations is taken into account by combining QM energies with average MM Poisson-Boltzmann free energies, resulting in a total free energy barrier of 14.8 kcal/mol in good agreement with experimental data. The rate-determining event in the MMP-2 mechanism corresponds to a H-bond rearrangement involving the Glu 404 residue and/or the Glu 404-COOH --> N-Ile( P 1') proton transfer. Overall, the present computational results and previous experimental data complement each other well in order to provide a detailed view of the MMPs catalytic mechanism.     

Proton transfer and associated molecular rearrangements in the photocycle of photoactive yellow protein: role of water molecular migration on the proton transfer reaction

The mechanism of the proton transfer and the concomitant molecular structural and hydrogen bond rearrangements after the photoisomerization of the chromophore in the photocycle of photoactive yellow protein are theoretically investigated by using the QM/MM method and molecular dynamics calculations. The free energy surface along this proton-transfer process is determined. This work suggests the important role of the water molecular migration into the moiety of chromophore, which facilitates proton transfer by the hydrogen bond rearrangement and the hydration of the pB' state.     

DNA polymerase beta catalysis: are different mechanisms possible?

DNA polymerases are crucial constituents of the complex cellular machinery for replicating and repairing DNA. Discerning mechanistic pathways of DNA polymerase on the atomic level is important for revealing the origin of fidelity discrimination. Mammalian DNA polymerase beta (pol beta), a small (39 kDa) member of the X-family, represents an excellent model system to investigate polymerase mechanisms. Here, we explore several feasible low-energy pathways of the nucleotide transfer reaction of pol beta for correct (according to Watson-Crick hydrogen bonding) G:C basepairing versus the incorrect G:G case within a consistent theoretical framework. We use mixed quantum mechanics/molecular mechanics (QM/MM) techniques in a constrained energy minimization protocol to effectively model not only the reactive core but also the influence of the rest of the enzymatic environment and explicit solvent on the reaction. The postulated pathways involve initial proton abstraction from the terminal DNA primer O3'H group, nucleophilic attack that extends the DNA primer chain, and elimination of pyrophosphate. In particular, we analyze several possible routes for the initial deprotonation step: (i) direct transfer to a phosphate oxygen O(Palpha) of the incoming nucleotide, (ii) direct transfer to an active site Asp group, and (iii) transfer to explicit water molecules. We find that the most probable initial step corresponds to step (iii), involving initial deprotonation to water, which is followed by proton migration to active site Asp residues, and finally to the leaving pyrophosphate group, with an activation energy of about 15 kcal/mol. We argue that initial deprotonation steps (i) and (ii) are less likely as they are at least 7 and 11 kcal/mol, respectively, higher in energy. Overall, the rate-determining step for both the correct and the incorrect nucleotide cases is the initial deprotonation in concert with nucleophilic attack at the phosphate center; however, the activation energy we obtain for the mismatched G:G case is 5 kcal/mol higher than that of the matched G:C complex, due to active site structural distortions. Taken together, our results support other reported mechanisms and help define a framework for interpreting nucleotide specificity differences across polymerase families, in terms of the concept of active site preorganization or the so-called "pre-chemistry avenue".     

Molecular dynamics and free energy perturbation study of hydride-ion transfer step in dihydrofolate reductase using combined quantum and molecular mechanical model

We used molecular dynamics simulation and free energy perturbation (FEP) methods to investigate the hydride-ion transfer step in the mechanism for the nicotinamide adenine dinucleotide phosphate (NADPH)-dependent reduction of a novel substrate by the enzyme dihydrofolate reductase (DHFR). The system is represented by a coupled quantum mechanical and molecular mechanical (QM/MM) model based on the AM1 semiempirical molecular orbital method for the reacting substrate and NADPH cofactor fragments, the AMBER force field for DHFR, and the TIP3P model for solvent water. The FEP calculations were performed for a number of choices for the QM system. The substrate, 8-methylpterin, was treated quantum mechanically in all the calculations, while the larger cofactor molecule was partitioned into various QM and MM regions with the addition of “link” atoms (F, CH₃, and H). Calculations were also carried out with the entire NADPH molecule treated by QM. The free energies of reaction and the net charges on the NADPH fragments were used to determine the most appropriate QM/MM model. The hydride-ion transfer was also carried out over several FEP pathways, and the QM and QM/MM component free energies thus calculated were found to be state functions (i.e., independent of pathway). A ca. 10 kcal/mol increase in free energy for the hydride-ion transfer with an activation barrier of ca. 30 kcal/mol was calculated. The increase in free energy on the hydride-ion transfer arose largely from the QM/MM component. Analysis of the QM/MM energy components suggests that, although a number of charged residues may contribute to the free energy change through long-range electrostatic interactions, the only interaction that can account for the 10 kcal/mol increase in free energy is the hydrogen bond between the carboxylate side chain of Glu30 (avian DHFR) and the activated (protonated) substrate. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 977–988, 1998     

Towards understanding phosphonoacetaldehyde hydrolase: an alternative mechanism involving proton transfer that triggers P-C bond cleavage

The theoretical QM/MM study of the reaction catalysed by phosphonoacetaldehyde hydrolase indicates a possible alternative mechanism of the P-C bond cleavage: as opposed to the mechanism proposed earlier and that involved formation of a covalently bound intermediate (Schiff-base), in the new mechanism, the bond breaking process is facilitated by proton transfer from catalytic lysine residue to the substrate.     

Antibiotic deactivation by a dizinc beta-lactamase: mechanistic insights from QM/MM and DFT studies

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods and density functional theory (DFT) were used to investigate the initial ring-opening step in the hydrolysis of moxalactam catalyzed by the dizinc L1 beta-lactamase from Stenotrophomonas maltophilia. Anchored at the enzyme active site via direct metal binding as suggested by a recent X-ray structure of an enzyme-product complex (Spencer, J.; et al. J. Am. Chem. Soc. 2005, 127, 14439), the substrate is well aligned with the nucleophilic hydroxide that bridges the two zinc ions. Both QM/MM and DFT results indicate that the addition of the hydroxide nucleophile to the carbonyl carbon in the substrate lactam ring leads to a metastable intermediate via a dominant nucleophilic addition barrier. The potential of mean force obtained by SCC-DFTB/MM simulations and corrected by DFT/MM calculations yields a reaction free energy barrier of 23.5 kcal/mol, in reasonable agreement with the experimental value of 18.5 kcal/mol derived from kcat of 0.15 s(-1). It is further shown that zinc-bound Asp120 plays an important role in aligning the nucleophile, but accepts the hydroxide proton only after the nucleophilic addition. The two zinc ions are found to participate intimately in the catalysis, consistent with the proposed mechanism. In particular, the Zn(1) ion is likely to serve as an "oxyanion hole" in stabilizing the carbonyl oxygen, while the Zn(2) ion acts as an electrophilic catalyst to stabilize the anionic nitrogen leaving group.     

Quantum mechanical/molecular mechanical study on the mechanisms of compound I formation in the catalytic cycle of chloroperoxidase: an overview on heme enzymes

The formation of Compound I (Cpd I), the active species of the enzyme chloroperoxidase (CPO), was studied using QM/MM calculation. Starting from the substrate complex with hydrogen peroxide, FeIII-HOOH, we examined two alternative mechanisms on the three lowest spin-state surfaces. The calculations showed that the preferred pathway involves heterolytic O-O cleavage that proceeds via the iron hydroperoxide species, i.e., Compound 0 (Cpd 0), on the doublet-state surface. This process is effectively concerted, with a barrier of 12.4 kcal/mol, and is catalyzed by protonation of the distal OH group of Cpd 0. By comparison, the path that involves a direct O-O cleavage from FeIII-HOOH is less favored. A proton coupled electron transfer (PCET) feature was found to play an important role in the mechanism nascent from Cpd 0. Initially, the O-O cleavage progresses in a homolytic sense, but as soon as the proton is transferred to the distal OH, it triggers an electron transfer from the heme-oxo moiety to form water and Cpd I. This study enables us to generalize the mechanisms of O-O activation, elucidated so far by QM/MM calculations, for other heme enzymes, e.g., cytochrome P450cam, horseradish peroxidase (HRP), nitric oxide synthase (NOS), and heme oxygenase (HO). Much like for CPO, in the cases of P450 and HRP, the PCET lowers the barrier below the purely homolytic cleavage alternative (in our case, the homolytic mechanism is calculated directly from FeIII-HOOH). By contrast, the absence of PCET in HO, along with the robust water cluster, prefers a homolytic cleavage mechanism.     

Phosphorylation reaction in cAPK protein kinase-free energy quantum mechanical/molecular mechanics simulations

We present results of a theoretical analysis of the phosphorylation reaction in cAMP-dependent protein kinase using a combined quantum mechanical and molecular mechanics (QM/MM) approach. Detailed analysis of the reaction pathway is provided using a novel QM/MM implementation of the nudged elastic band method, finite temperature fluctuations of the protein environment are taken into account using free energy calculations, and an analysis of hydrogen bond interactions is performed on the basis of calculated frequency shifts. The late transfer of the substrate proton to the conserved aspartate (D166), the activation free energy of 15 kcal/mol, and the slight exothermic (-3 kcal/mol) character of the reaction are all consistent with the experimental data. The near attack conformation of D166 in the reactant state is maintained by interactions with threonine-201, asparagine-177, and most notably by a conserved water molecule serving as a strong structural link between the primary metal ion and the D166. The secondary Mg ion acts as a Lewis acid, attacking the beta-gamma bridging oxygen of ATP. This interaction, along with a strong hydrogen bond between the D166 and the substrate, contributes to the stabilization of the transition state. Lys-168 maintains a hydrogen bond to a transferring phosphoryl group throughout a reaction process. This interaction increases in the product state and contributes to its stabilization.     

H-ZSM-5分子筛上环己烯芳构化反应历程的理论研究

基于76T簇模型,采用量子力学和分子力学联合的ONIOM2(B3LYP/6-31G(d,p):UFF)方法研究了H-ZSM-5分子筛上环己烯芳构化反应历程.结果表明,环己烯首先吸附在分子筛酸性位上,与酸性质子共同脱除一个H2分子后,在分子筛骨架氧上生成烷氧配合物中间体;然后再脱质子得到环己二烯,同时酸性位复原;再经历脱氢和脱质子历程,最后得到产物苯,并吸附在复原的分子筛酸性位上.计算得到脱氢的活化能依次为279.64和260.21kJ/mol,脱质子的活化能依次为74.64和59.14kJ/mol.所有脱氢反应都是吸热过程,生成表面烷氧活性中间体,随后的脱质子反应能垒较低,而且是放热过程.此外,比较了环己烯在分子筛酸性位上的三个竞争反应,即脱氢、质子化和氢交换反应的活化能垒,证明环己烯优先发生脱氢反应.     

QM/MM study of mechanisms for compound I formation in the catalytic cycle of cytochrome P450cam

In the catalytic cycle of cytochrome P450cam, after molecular oxygen binds as a ligand to the heme iron atom to yield a ferrous dioxygen complex, there are fast proton transfers that lead to the formation of the active species, Compound I (Cpd I), which are not well understood because they occur so rapidly. In the present work, the conversion of the ferric hydroperoxo complex (Cpd 0) to Cpd I has been investigated by combined quantum-mechanical/molecular-mechanical (QM/MM) calculations. The residues Asp(251) and Glu(366) are considered as proton sources. In mechanism I, a proton is transported to the distal oxygen atom of the hydroperoxo group via a hydrogen bonding network to form protonated Cpd 0 (prot-Cpd0: FeOOH(2)), followed by heterolytic O-O bond cleavage that generates Cpd I and water. Although a local minimum is found for prot-Cpd0 in the Glu(366) channel, it is very high in energy (more than 20 kcal/mol above Cpd 0) and the barriers for its decay are only 3-4 kcal/mol (both toward Cpd 0 and Cpd I). In mechanism II, an initial O-O bond cleavage followed by a concomitant proton and electron transfer yields Cpd I and water. The rate-limiting step in mechanism II is O-O cleavage with a barrier of about 13-14 kcal/mol. According to the QM/MM calculations, the favored low-energy pathway to Cpd I is provided by mechanism II in the Asp(251) channel. Cpd 0 and Cpd I are of similar energies, with a slight preference for Cpd I.     

Hybrid QM/MM and DFT investigations of the catalytic mechanism and inhibition of the dinuclear zinc metallo-beta-lactamase CcrA from Bacteroides fragilis

Based on hybrid QM/MM molecular dynamics simulation and density functional theoretical (DFT) calculations, we investigate the mechanistic and energetic features of the catalytic action of dizinc metallo-beta-lactamase CcrA from Bacteroides fragilis. The 200 ps QM/MM simulation of the CcrA enzyme in complex with nitrocefin shows that the substrate beta-lactam moiety is directed toward the active site dizinc center through the interactions of aminocarbonyl and carboxylate groups with the two active site zinc ions and the two conserved residues, Lys167 and Asn176. From the determination of the potential energy profile of a relevant enzymatic reaction model, it is found that the nucleophilic displacement reaction step proceeds with a low-barrier height, leading to the formation of an energetically favored reaction intermediate. The results also show that the high catalytic activity of the CcrA enzyme stems from a simultaneous operation of three catalytic components: activation of the bridging hydroxide nucleophile by zinc-coordinated Asp86; polarization of the substrate aminocarbonyl group by the first zinc ion; stabilization of the negative charge developed on the departing amide nitrogen by the second zinc ion. Consistent with the previous experimental finding that the proton-transfer reaction step is rate-limiting, the activation energy of the second step is found to be 1.6 kcal/mol higher than that of the first step. Finally, through an examination of the structural and energetic features of binding of a thiazolidinecarboxylic acid inhibitor to the active site dizinc center, a two-step inhibition mechanism involving a protonation-induced ligand exchange reaction is proposed for the inhibitory action of a tight-binding inhibitor possessing a thiol group.     

Quantum mechanics/molecular mechanics minimum free-energy path for accurate reaction energetics in solution and enzymes: sequential sampling and optimization on the potential of mean force surface

To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations of the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical SN2 reaction of Cl-+CH3Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.     

A quantum mechanical/molecular mechanical study on the catalysis of the pyridoxal 5'-phosphate-dependent enzyme L-serine dehydratase

The catalytic mechanism of a pyridoxal 5'-phosphate-dependent enzyme, l-serine dehydratase, has been investigated using ab initio quantum mechanical/molecular mechanical (QM/MM) methods. New insights into the chemical steps have been obtained, including the chemical role of the substrate carboxyl group in the Schiff base formation step and a proton-relaying mechanism involving the phosphate of the cofactor in the beta-hydroxyl-leaving step. The latter step is of no barrier and follows sequentially after the elimination of the alpha-proton, leading to a single but sequential alpha, beta-elimination step. The rate-limiting transition state is specifically stabilized by the enzyme environment. At this transition state, charges are localized on the substrate carboxyl group, as well as on the amino group of Lys41. Specific interactions of the enzyme environment with these groups are able to lower the activation barrier significantly. One major difficulty associated with studies of complicated enzymatic reactions using ab initio QM/MM models is the appropriate choices of reaction coordinates. In this study, we have made use of efficient semiempirical models and pathway optimization techniques to overcome this difficulty.