Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the...

download Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

of 199

Transcript of Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the...

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    1/199

    Proefschrift voorgelegd tot het behalen van de wettelijke graad van Doctor in de Wetenschappen

    Promotoren: Prof. Dr. Paul GeerlingsProf. Dr. Lode WynsDr. ir. Joris Messens

    Goedele Roos

    Mei 2007

    Theory meets experiment:a combined quantum chemical-experimental study

    of the reaction mechanism of pI258 arsenate reductase

    H

    y=Ey

    Hy=Ey

    VrijeUn

    iversiteitBrussel

    FaculteitWetenschappen

    Onderzo

    eksgroepAlgemenechemie

    LaboratoriumvoorUltrastructuur

    VIBDep

    artementMoleculaireenCellulaireInteracties

    a-helixa-helix

    Cys82Cys82redoxhelixredoxhelix

    Cys89Cys89

    Cys10Cys10

    P-loopP-loop

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    2/199

    Promotoren: Prof. Dr. Paul GeerlingsProf. Dr. Lode WynsDr. ir. Joris Messens

    Goedele Roos

    Mei 2007

    Theory meets experiment:

    a combined quantum chemical-experimental study

    of the reaction mechanism of pI258 arsenate reductase

    VrijeUniversiteitBrussel

    Faculteit

    Wetenschappen

    Onderzo

    eksgroepAlgemenechemie

    Laborato

    riumvoorUltrastructuur

    VIBDep

    artementMoleculaireenCellulaireInteracties

    Proefschrift voorgelegd tot het behalen van de wettelijke graad van Doctor in de Wetenschappen

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    3/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    4/199

    Thank you

    Thank you for breaking my heart

    Thank you for breaking me apart

    Now Ive a strong, strong heart

    Thank you for breaking my heart

    (Sinad OConnor)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    5/199

    ii

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    6/199

    iii

    Im sincerely grateful to

    Prof. Dr. Paul Geerlings, Prof. Dr. Lode Wyns and Dr. ir. Joris Messens, for supervising thiswork.Prof. Dr. ir. Remy Loris, for teaching me the fundamentals of crystallography.Prof. Dr. Frank De Proft, Dr. ir. Stefan Loverix, Dr. ir. Lieven Buts and Abel Garcia-Pino, for

    fascinating scientific discussions.

    Mr. Jan Moens and Ms. Lies Broeckaert, the students I supervise.Ms. Elke Brosens, ir. Khadija Wahni and Ms. Karolien Van Belle, for outstanding experimental

    work.

    Mr. Wim Cossement, for magic help with IT problems.Ms. Maria Vanderveken, Ms. Nadine Desmaels and Ms. Martine Vandeperre, for administrative

    help.

    Mr. Bruno Janssens, for technical support.

    Ms. Diane Sorgeloos, for pleasant collaboration during student lab classes.All people from ALGC and ULTR, for useful discussions and collegiality.The VUB/ULB computer centre and the FWO, for computation time and financial support.Ms. Evelyne Namenwirth and Dr. Wim Vandendooren, for hearing me.Bonneke & Bompa, for not giving up on me.Julianna and Christa, my Dear Friends..and my Dearest Friend for his never ending TLC

    Thank you

    Goedele

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    7/199

    iv

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    8/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    9/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    10/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    11/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    12/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    13/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    14/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    15/199

    xii

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    16/199

    CHAPTER I

    An Introduction

    Something unknown is doing we don't know what.

    (Sir Arthur Eddington)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    17/199

    2

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    18/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    19/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    20/199

    Chapter I: Introduction

    5

    light of experimental data. As such, our work fits in a multidisciplinary approach, combining theoretical

    and experimental studies to gain full insight into the enzymatic reaction mechanism of pI258 ArsC.

    3. OutlineAfter this general introduction, Chapter II gives an overview of the biochemical and structural

    characteristics of pI258 arsenate reductase from Staphylococcus aureus. In Chapter III, the

    fundamentals of quantum chemistry are discussed, with special attention to Density Functional Theory.

    In Chapter IV we focus on the phosphatase-like nucleophilic displacement reaction carried out by a

    nucleophilic cysteine on arsenate, leading to a covalent enzyme-arseno adduct. In Chapter VII the

    nucleophilic attack on this enzyme-arseno adduct and the looping-out of an -helix is studied. After one

    catalytic cycle, ArsC is in its oxidized form and needs to be regenerated to its reduced form bythioredoxin. In Chapter IX, the reduction power of thioredoxin is discussed. In Chapter VIII the special

    feature of potassium binding in pI258 ArsC is treated.

    Chapter V and VI are intermezzi, handling more fundamental work on the metastable nature of multiply

    charged anions and on the origin of the pKa perturbation at the N-terminal of helices, respectively.

    While I was doing the quantum chemical parts of this thesis, the experiments reported in Chapter VII

    and VIII were designed and coordinated by Joris Messens and carried out by Lieven Buts (ITC

    measurements), Karolien Van Belle (kinetic studies), Elke Brosens (construction of ArsC mutants),

    Remy Loris (crystallographic data collection and solving the ArsC structures) and Joris Messens(solving the ArsC structures). I performed parts of the experimental work of Chapter IX (pKa

    measurements, chemical unfolding, solving the thioredoxin structures) under supervision of Joris

    Messens, who designed and coordinated the experiments, and Remy Loris, who taught me how to solve

    the thioredoxin crystal structures. For the rest of the experimental work, credits go to Abel Garcia-Pino

    (DSC measurements), Elke Brosens (construction of the thioredoxin mutants and seeding experiments),

    Karolien Van Belle (kinetic studies and redox potential measurements), Remy Loris (crystallographic

    data collection) and Guy Vandenbussche (mass spectrometry).

    In Chapter VIII and IX, the results and discussion section are separated to improve readability.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    21/199

    6

    References

    1. Healy, S. M., Wildfang, E., Zakharyan, R. A., Aposhian, H. V., Biol. Trace Elem. Res.1999, 68, 249.2. Rensing, C., Ghosh, M., Rosen, B. P., J. Bacteriol.1999, 181, 5891.3. Dey, S., Rosen, B. P., J. Bacteriol.1995, 177, 385.4. Messens, J., Martin, J. C., Van Belle, K., Brosens, E., Desmyter, A., De Gieter, M., Wieruszeski, J.-M., Willem, R., Wyns, L.,

    Zegers, I., Proc. Natl. Acad. Sci. USA2002, 99, 8506.5. Ji, G., Silver, S., Proc. Natl Acad. Sci. USA1992, 89, 9474.6. Ji, G., Garber, E. A., Armes, L. G., Chen, C. M., Fuchs, J. A., Silver, S., Biochemistry1994, 33, 7294.7. Messens, J., Martins, J. C., Brosens, E., Van Belle, K., Jacobs, D. M., Willem, R., Wyns, L., J. Biol. Inorg. Chem. 2002, 7, 146.8. Gladysheva, T. B., Oden, K. L., Rosen, B. P., Biochemistry1994, 33, 7288.9. Shi, J., Vlamis-Gardikas, A., Aslund, F., Holmgren, A., Rosen, B. P., J. Biol. Chem.1999, 274, 36039.10.

    Mukhopadhyay, R., Rosen, B. P., FEMS Microbiol. Lett. 1998, 168, 127.11. Mukhopadhyay, R., Shi, J., Rosen, B. P., J. Biol. Chem.2000, 275, 21149.

    12. Challenger, F., Chem. Rev. 1945, 36, 315.13. Messens, J., Silver, S., J. Mol. Biol. 2006, 362, 1.14. Stolz, J. F., Oremland, R. S., FEMS Microbiol. Rev.1999, 23, 615.15. Herschlag, D., Jencks, W. P., J. Am. Chem. Soc.1989, 111, 7587.16. Hengge, A. C., Cleland, W. W., J. Am. Chem. Soc.1990, 112, 7421.17. Nray-Szab, G., Theochem.2000, 500, 157.18. Rawlings, N. D., Polgr, L., Barrett, A. J., Biochem. J. 1991, 279, 907.19.

    Baeten, A., Maes, D., Geerlings, P., J. Theoret. Biol.

    1998, 195, 2711.20. Mignon, P., Steyaert, J., Loris, R., Geerlings, P., Loverix, S., J. Biol. Chem.2002, 39, 36770.

    21. Versees, W., Loverix S., Vandemeulebroeke A., Geerlings P., Steyaert, J., J. Mol. Biol.2004, 338, 1.22. Johannin, G., kellersohn, N., Biochem. Biophys. Res. Commun. 1972, 49, 321.23. Lagunas, R., Pestana, D., Diez-Masa, J. C., Biochemistry1984, 5, 955.24. Hohenberg, P., Kohn, W., Phys. Rev.1960, B136, 864.25. Parr R. G., Yang W., Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989.26. Geerlings, P., De Proft, F., Langenaeker, W., Chem. Rev,2003, 103, 1793.27. Pearson, R. G., Parr, R. G., J. Am. Chem. Soc.1983, 105, 7512.28. Chattaraj, P. K., Lee, H., Parr, R. G., J. Am. Chem. Soc. 1991, 113, 1855.29. Gzquez, J. L., Mendez, F. J., J. Am. Chem. Soc. 1994, 98, 4591.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    22/199

    Chapter IIBiochemical and structural characteristics

    of pI258 arsenate reductasefrom Staphylococcus aureus

    Most of the fundamental ideas of science are essentially simple,

    and may, as a rule,

    be expressed in a language comprehensible to everyone.

    (Einstein)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    23/199

    8

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    24/199

    Chapter II: pI258 ArsC

    9

    Arsenate reductase (ArsC) from Staphylococcus aureus plasmid pI258, a 14.8 kDa monomeric protein,plays a role in bacterial heavy metal resistance and catalyzes the reduction of arsenate to arsenite. ArsCis part of the ars operon coding for ArsR and ArsB in addition to ArsC1. ArsR is a regulatory proteinrepressing protein transcription in response to arsenite2. ArsB is a proton-driven transport system thatextrudes arsenite3.

    1. ArsC has a PTPase I fold

    Despite the low sequence identity (< 20 %) ofS.aureus ArsC with low molecular weight phosphatase(LMW PTPase), ArsC has the characteristic PTPase I fold: a four stranded parallel -sheet and threemajor-helices (Fig. 2.1A)4. Arsenate reduction is the third function associated with a PTPase I fold

    after tyrosine dephosphorylation

    5

    and cellobiose phosphorylation

    6

    . The catalytic site of LMW PTPase isconserved in ArsC. In LMW PTPase, this site is composed of the oxyanion binding loop including thenucleophilic Cys13, the conserved Asn16 and Arg19 (numbering in LMW PTPase of Saccharomycescerevisiae7), called the P-loop. In ArsC, the equivalent residues are Cys10, Asn13 and Arg16 (Fig.2.1B)4. The Tyr or Ser residues lining the binding site in PTPase are conserved in ArsC (Ser17)4.

    Figure 2.1: The structure of pI258 ArsC.A. Overall structure of the reduced form of arsenate reductase.

    The P-loop is shown in red and the catalytic important -helices are shown in yellow. B. Oxyanion bindingP-loop including the conserved residues Cys10, Asn13, Arg16 and Ser17. The figure was generated byusing PyMol (Delano Scientific LLC 2005) from the PDB coordinates of 1LJL.

    A B

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    25/199

    10

    During arsenate reduction and dephosphorylation, a water or tyrosine molecule is split off the substrate.The overall geometry adopted by the P-loop of ArsC in the first reaction step is different from that ofPTPases, possibly because the smaller substrate permits a different orientation of the leaving group8. In

    ArsC, the leaving water is much more buried in the active site. It is cradled by N and N of Arg16 andclose to (2.6 ) a water molecule that in turn is hydrogen bonded to Asp105 (Fig. 2.2) 8. In LMWPTPase, strong hydrogen bonds are formed between NH/NH of the guanidinium group of the ArsCArg16 homologue (Arg19 in PTPase of S. cerevisiae7) and two non-protonated oxygen atoms of the

    phosphotyrosine substrate9. The equivalent of Asp105 in LMW PTPase (Asp132 in PTPase of S.cerevisiae7) protonates the leaving oxygen in the dephosphorylation reaction9. In ArsC, mutatingAsp105 to alanine (KM = 3.8 mM, kcat = 58.5 min

    -1)8 increases the KM with a factor of 55 and decreasesits kcat about four times compared to wild type ArsC (KM = 68 M, kcat = 215 min

    -1)10. The respecticeAsp/Ala mutation in LMW PTPase, however, decreases the kcat with a factor of more than 1,000, whilehardly affecting KM

    11,12. Therefore, in ArsC, Asp105 might have a somewhat different function,stabilizing the transition state via a bound (protonated) water molecule8.

    Figure 2.2: Orientation of the leaving group in the P-loop of ArsC (A) and LMW PTPase (B). Thefigure was generated by using PyMol (Delano scientific LLC 2005) from the PDB coordinates of 1LJU(product structure of pI258 ArsC) and 1D1P (LMW PTPase of S. cerevisiae in complex with 4-(2-hydroxyethyl)-1-piperazine ethanesulfonic acid represented as methanesulfonic acid as substrateanalogue).

    A B

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    26/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    27/199

    12

    Figure 2.3: Scheme of the reaction mechanism of pI258 ArsC. 1. The reaction starts with thenucleophilic attack of Cys10 on arsenate leading to a covalent intermediate. 2. Arsenite is released afterthe nucleophilic attack of the thiol of Cys82. A Cys10-Cys82 intermediate is formed and the redox helixpartially unfolds. 3. At the end of the reduction cycle, Cys89 attacks Cys82, forming a Cys82-Cys89disulfide. The redox helix is looped-out and presents the disulphide bridge at the surface of the enzyme tothioredoxin. 4. Thioredoxin (Trx) regenerates the reduced form of arsenate reductase for a subsequent

    catalytic cycle. The figure was generated by using PyMol (Delano Scientific LLC 2005).

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    28/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    29/199

    14

    The experimental determination of the pKa of the catalytic important thiol groups (Cys 10, Cys82 andCys89) is not straightforward because these redox active cysteine residues are all involved in thesuccessive steps of the reaction mechanism4 (Fig. 2.3). At pH 8.0, free cysteine (pKa = 8.3) is largely

    present in the thiol form, which is a far inferior nucleophile than the thiolate form20. However, theacid/base properties of functional groups may be perturbed in a protein environment as compared toaqueous solution21. In addition to the nature of the nucleophile, the pre-organized environment of anenzyme might alter the leaving group as compared to the cases where these entities are isolated in gas

    phase or in solution21. Analysis of the interactions in the ArsC-substrate complex (Chapter IV) and inthe ArsC-arseno covalent adduct (Chapter VII) provides insight into the structural features of ArsCrelated to its capability to activate both the leaving groups (water and arsenite) and the nucleophiles(Cys10, Cys82 and Cys89) in the reactant state.

    For its activity, pI258 ArsC benefits from the binding of tetrahedral oxyanions in the P-loop active siteand from the binding of potassium in a specific cation-binding site. Further, in the P-loop the peptide

    bond between Gly12 and Asn13 can adopt two distinct conformations. These special features of potassium binding and - flipping in pI258 ArsC together with the tetrahedral-anion-dependentcatalysis in pI258 ArsC are studied (Chapter VIII).

    Thioredoxin (Trx) regenerates the reduced form of ArsC for a subsequent catalytic cycle. What makesTrx a reducing agent? This question is answered in Chapter IX. The presence of a highly conserved

    proline in the WCGPC active site motif of Trx drew our attention. The role of this proline residue in thereducing force of Trx is unravelled.

    Chapter II: pI258 ArsC

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    30/199

    Chapter II: pI258 ArsC

    15

    References

    1. Ji, G., Silver, S., Proc. Natl. Acad. Sci. USA1992, 89, 9474.2.

    Silver, S., Plasmid1992, 27, 1.3. Brer, S., Ji, G., Brer, A., Silver, S., J. Bacteriol1993, 175, 3840.

    4. Zegers, I., Martins, J. C., Willem, R.,Wyns, L., Messens, J., Nature Struct. Biol.2001, 8, 843.5. Denu, J. M., Dixon, J. E., Curr. Opin. Chem. Biol. 1998, 2, 633.6. Ab, E., Schuurman-Wolters, G., Reizer, J., Saier, M. H., Dijkstra, K., Scheek, R. M., Robillard, G. T., Protein Sci.1997, 6, 304.7. Wang, S., Tabernero, L., Zhang, M., Harms, E., Van Etten, R., Stauffacher, C. V., Biochemistry 2000, 39, 1903.8. Messens, J., Martins, J. C., Van Belle, K., Brosens, E., Desmyter, A., De Gieter, M., Wieruszeski, J-M., Willem, R., Wyns, L.,

    Zegers, I., Proc. Natl. Acad. Sci USA2002, 99, 8506.9. Zhang, Z.-Y., Critical Reviews in Biochemistry and Molecular Biology1998, 33, 1.10. Messens, J., Martins, J. C., Brosens, E., Van Belle, K., Jacobs, D. M., Willem, R., Wyns, L., J. Biol. Inorg. Chem.2002, 7, 146.11. Wu, L., Zhang, Z. Y., Biochemistry 1996, 35, 5426.12. Taddei, N., Chiarugi, P. Cirri, P., Fiaschi, T., Stefani, M., Camici, G., Raugei, G., Ramponi, G., FEBS Lett.1994, 350, 328.13. Ji, G., Garber, E. A., Armes, L. G., Chen, C. M., Fuchs, J. A., Silver, S., Biochemistry1994, 33, 7294.14. Selwyn, M. J., Biochim. Biophys. Acta1965, 105, 193.15. Jacobs, D. M., Messens, J., Wechselberger, R. W., Brosens, E., Willem, R., Wyns, L., Martins, J. C., J. Biomol. NMR2001, 20,

    95.16. Ramponi, G., Stefani, M., Biochim. Biophys. Acta 1997, 1341, 137.17. Stolz, J. F., Oremland, R. S., FEMS Microbiol. Rev.1999, 23, 615.18. Messens, J., Hayburn, G., Desmyter, A., Laus, G., Wyns, L., Biochemistry1999, 38, 16857.19. Messens, J., Van Molle, I., Vanhaesebrouck, P., Limbourg, M., Van Belle, K., Wahni, K., Martins, J. C., Loris, R., Wyns, L., J.

    Mol. Biol.2004, 339, 527.20. Dantzman, C. L., Kiessling, L. L., J. Am. Chem. Soc.1997, 118, 11715.21. Fersht, A., Enzyme Structure and Mechanism, W. H. Freeman and Company, New York, 1984.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    31/199

    16

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    32/199

    Chapter IIITheoretical background

    Any one who is not shocked by quantum mechanics has not fully understood it.

    (Niels Bohr)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    33/199

    18

    Chapter III: Theoretical background

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    34/199

    p g

    19

    1. Fundamentals

    Quantum chemistry is based on an approximate solution of Schrodinger's time independent equation1,2

    from which all electronic properties of atoms and molecules can be derived:

    = EH (3.1)

    in whichHis the Hamilton operator for a system of electrons and nuclei, is the wave function andEis the energy. For a system constituted of Mnuclei and N electrons, the non-relativistic Hamiltonian,

    written in atomic units (a.u.), is given by:

    =

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    35/199

    Chapter III: Theoretical background

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    36/199

    21

    2. Hartree-Fock theory

    2.1 The Slater determinantThe wave function must satisfy the anti-symmetry principle (the Pauli exclusion principle), which states

    that a wave function must change sign when the spatial and spin components of any two electrons are

    exchanged. In the Hartree-Fock scheme, the simplest possible anti-symmetric wave function (i. e. a

    single determinant) is used to describe the ground state of anN-electron system. This single determinant

    wave function is the Slater determinant:

    )()()(

    )2()2()2(

    )1()1()1(

    !

    1

    21

    21

    21

    0

    NNN

    N

    N

    N

    N

    SD

    K

    KKKK

    K

    K

    =

    (3.6)

    with!

    1

    Nthe normalization factor and )(ji the molecular spin orbitals depending on three spatial

    coordinates of electronj and one spin coordinate:

    ),()()( jjijii rxj == (3.7)

    To a very good approximation, the Hamiltonian in eq. 3.4 does not involve the spin variables, but is

    only a function of the spatial coordinates. Consequently, the molecular spin orbitals can be written as

    the product of a spatial orbital )( ji r and a spin function )( j or )( j , corresponding to a spin

    up or a spin down situation:

    =

    )()()()(

    j

    jjii rj

    (3.8)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    37/199

    22

    A molecular orbital is defined as a mono-electronic wave function characterizing an electron in a

    molecular system. It can be expanded in a set ofK basis functions { } , the atomic orbitals, withexpansion coefficients ci:

    =

    =K

    ii cj1

    )(

    (3.9)

    2.2 The variational methodHartree-Fock theory is based on a variational procedure11. If is any anti-symmetric normalized

    function of the electronic coordinates, then the energy associated to this function is:

    dHE elec= * (3.10)

    in which the integration is over all the coordinates. If is the exact wave function of the ground state,Ewillbe the exact energy of the ground state (E0).However, if is any normalized anti-symmetricwave function different from the exact ground state wave function, the associated energyEislarger than

    the exact ground state energy (E> E0). This calls for a variational method in which the parameters of the

    wave function should be varied until the energy associated to the wave function is minimal. Thevariational method can be applied to determine the optimum orbitals of a single determinant wave

    function. The coefficients ci in eq. 3.9 must be adjusted to minimize the energy, implying:

    0=

    ic

    E

    (3.11)

    Chapter III: Theoretical background

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    38/199

    23

    2.3 Closed shell systemsStudies of closed shell systems (no unpaired electrons) can be performed in a restricted Hartree-Fock

    (RHF) calculation12 in which each orbital has two electrons, one spin up, the other spin down. In theassumption of identical molecular orbitals for and electrons, the variational condition (eq. 3.11)

    leads to a set of algebraic equations forci, the so-called Hartree-Fock Roothaan-Hall equations12,13:

    0)(1

    ==

    i

    N

    i cSF

    = 1,2,,N (3.12)

    fulfilling the normalization condition:

    11 1

    * == =

    i

    N N

    i cSc

    (3.13)

    In this equation, i represents the one-electron energy of the molecular orbital i ; S are the

    elements of the overlap matrix:

    1* )1()1( rdS = (3.14)

    and F the elements of the Fock matrix:

    F = Hcore + P ( )

    1

    2 ( )

    =1

    K

    =1

    K

    (3.15)

    The matrix elementscoreH are associated to the mono-electronic Hamiltonian describing the kinetic

    energy of electrons and the electron-nuclei attraction (see eq. 3.2). It can be written as:

    1* )1()1()1( rdHH corecore = (3.16)

    with

    =

    =M

    A A

    Acore

    R

    ZH

    1 1

    2

    2

    1)1( (3.17)

    ZA is the atomic number of atomA andR1Ais the distance from electron 1 to atomA.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    39/199

    24

    The quantities (|) in the Fock matrix are the two-electron repulsion integrals:

    ( ) = *

    (r1) (r1)1

    r12

    *(r2 )(r2 )dr1dr2

    (3.18)

    These integrals are multiplied by the elements of the one-electron density matrixP:

    P = 2 ci*

    cii =1

    occ

    (3.19)

    in which the summation is over all occupied molecular orbitals. The factor of two indicates that each

    orbital is occupied by two electrons. One finally gets the total electronic energy:

    Eel =

    1

    2P(H

    core +F)=1

    K

    =1

    K

    (3.20)

    2.4 Open shell systemsIn the case of systems with an odd number of electrons, the electrons cannot be assigned in pairs to

    molecular orbitals. The molecular orbital theory commonly used for open shell systems is the spin-unrestricted Hartree-Fock (UHF) theory14. In this case different molecular orbitals are considered for the

    andelectrons, i.e. two sets of molecular orbitals are defined with two sets of coefficients:

    i = ci

    =1

    K

    (3.21)

    i

    = ci

    =1

    K

    (3.22)

    Chapter III: Theoretical background

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    40/199

    25

    The coefficients

    ic and

    ic are varied separately, leading to the Pople-Nesbet equations:

    0)(

    1

    ==

    i

    K

    i cSF = 1, ,K (3.23)

    0)(1

    ==

    i

    K

    i cSF = 1, ,K (3.24)

    In the open-shell case, the Fock matrices are defined as:

    F = H

    core + [(

    =

    1

    K

    =1

    K

    P +P )() P ( )]

    (3.25)

    and

    F = H

    core + [(=1

    K

    =1

    K

    P +P )() P ( )]

    (3.26)

    with the expressions for the density matrices:

    P = ci*cii=1

    ,occ

    (3.27)

    and

    P = ci

    *ci

    i=1

    ,occ

    (3.28)

    Finally, the electronic energy becomes:

    Eel =

    1

    2[(P

    +P

    )Hcore +

    =1

    K

    =1

    K

    PF +P F ]

    (3.29)

    l i f h k i

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    41/199

    26

    2.5 Solution of the Hartree-Fock equationsThe Hartree-Fock Roothaan-Hall or Pople-Nesbet equations determine the molecular orbital

    coefficients, together with the molecular orbital energies. However, the Fock matrix itself depends onthe molecular orbital coefficients. As such, the solution necessarily involves an iterative process. Since

    the molecular orbitals are derived from their own effective potential, the technique is called self-

    consistent-field (SCF) theory. Equation (3.12)can be written as a matrix equation:

    [ ][ ] [ ][ ] iii cScF = (3.30)

    with [F] the Fock matrix, [ci] the column matrix containing the coefficients ci of orbital i, [S] denoting

    the overlap integral matrix. Before the equations can be solved, they have to be transformed into a set of

    pseudo-eigenvalue equations. After the orthogonalisation of the orbitals we obtain:

    [ ][ ] [ ][ ] iii cScF '' = (3.31)

    The SCF procedure starts with an initial guess for the molecular orbital coefficients ci (associated with

    the density matrixP0). From this guess, the Fock matrix is calculated and diagonalized, giving a new set

    of molecular orbital coefficients associated with the density matrix P. From this, the Fock matrix is

    again constructed, repeating the above procedure. This is continued until the set of coefficients used to

    construct the Fock matrix is equal to those resulting from the diagonalization.

    P0

    F Ck P

    (3.32)

    3. Density Functional Theory

    3.1 IntroductionThe electronic wave function of anN-electron molecule depends on 3N spatial coordinates and on Nspin coordinates. This has prompted the search for functions that can be used to calculate energies and

    molecular properties involving fewer variables. The Density Functional Theory (DFT) based on the

    Hohenberg-Kohn theorems, uses the electron density(r) as the ground function containing physically

    significant information15-18

    . While the complexity of the wave function increases with the number of

    Chapter III: Theoretical background

    l h l d i i l d di h i l di i d d f h b

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    42/199

    27

    electrons, the electron density is only depending on three spatial coordinates independent of the number

    of electrons. In addition, several concepts which are known in chemistry, such as electronegativity and

    hardness, find a theoretical foundation in this theory.

    3.2 The Hohenberg-Kohn theoremsFor an electronic system, the ground state energy and wave function are determined by the minimization

    of the energy as the expectation value of the Hamiltonian (eq. 3.11). However for an N-electron system,

    the external potential (r) completely determines this Hamiltonian. As such, Nand (r)determine all

    properties of the ground state. Since(r) determines N ( =Nrdr)( ), the first Hohenberg-Kohntheorem legitimizes the use of the electron density(r) as basic variable. It states: The external potential

    (r) is determined, within a trivial additive constant, by the electron density(r)19.

    { }0 0 0, ,A AN Z R H E (3.33)

    The external potential (r) is the potential due to the nuclei of the molecular system. It is the classical

    nucleus-electron attraction and can be written at a position rfor a system ofMnuclei as:

    = =

    M

    A A

    A

    Rr

    Zr

    1

    )( (3.34)

    withRAandZA the position and charge of nucleusA respectively.

    The second Hohenberg-Kohn theorem provides the energy variational principle. It states: For a trial

    density )(' r with 0)(' r for all r and Nrdr = )(' is [ ])('0 rEE , with E0 the exact groundstate energy.

    The energy functional may be divided into several contributions. Since )(r determines all properties ofthe ground state, these should all be functionals of(r):

    [ ] [ ] [ ] [ ]( ) ( ) ( ) ( )ne eeE r T r V r V r = + + (3.35)

    in which [ ])(rT expresses the kinetic energy of the electrons, [ ])(rVne the nucleus-electron attractionenergy and [ ])(rVee the electron-electron repulsion energy.

    Th i f [ ])(E b i

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    43/199

    28

    The expression for [ ])(rE can be rewritten as:

    [ ] [ ] += rdrrrFrE HK )()()()( (3.36)

    with

    [ ] [ ] [ ])()()( rVrTrF eeHK += (3.37)

    and

    [ ] = rdrrrVne )()()( (3.38)

    The exact form of [ ])(rT for a system of interacting entities is unknown until now, but in the Kohn-Shame approach it is approximated by the kinetic energy for a non-interacting system.

    In analogy with the Hartree-Fock theory, the [ ])(rVee term may be divided into a Coulomb and anexchange part, [ ])(rJ and [ ])(rK implicitly including correlation energy:

    [ ] [ ] [ ]( ) ( ) ( )eeV r J r K r = + (3.39)

    with:

    [ ] 212112

    )()(1

    )( rdrdrrr

    rJ = (3.40)

    and [ ])(rK an unknown non-classical term. The nuclear-nuclear repulsion being constant in the Born-Oppenheimer approximation is omitted. The search for the ground state electron density )(r starts

    with the minimization condition:

    [ ] 0)()( =

    rrE

    (3.41)

    together with the constraint that the electron density should integrate to the total number of electronsN

    present in the atomic or molecular system:

    [ ] == rdrrNN )()( (3.42)

    Chapter III: Theoretical background

    This leads to the following Euler equation:

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    44/199

    29

    This leads to the following Euler equation:

    [ ] 0))()(()(

    = rrEr

    (3.43)

    In which is a Lagrange multiplier.

    Combination of eq. 3.36 and eq. 3.43 leads to:

    [ ]

    =+ ))()()(()( rdrrrFr HK

    (3.44)

    and, finally to:

    [ ]

    =+)(

    )()(r

    rFr HK (3.45)

    This equation can be considered as the density functional analogue of the Schrodinger equation. It can

    be used to determine the ground state electron density )(r .

    3.3 The Kohn-Sham methodEarlier attempts to deduce functionals for the kinetic and exchange energies considered a non-

    interacting uniform gas such as the Thomas-Fermi Dirac (TFD) model dating from the 1920s20.

    However, in this model the approximate forms for [ ])(rT and [ ])(rVee do not hold very well foratomic and molecular systems. In the model of a non-interacting uniform electron gas, the TFD theory

    does not predict bonding; molecules simply do not exist in this approach.

    The foundation for the use of DFT methods was the introduction of orbitals by Kohn and Sham21. In

    terms of these orbitals, the electron density becomes:

    drrN

    i

    i

    2

    ),()( =

    (3.46)

    The unknown kinetic energy functional [ ])(rT can consequently be written as:

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    45/199

    30

    The unknown kinetic energy functional [ ])(rT can consequently be written as:

    [ ] =i

    iiiinrT 2

    2

    1)(

    (3.47)

    with i and in the natural spin orbitals and occupation numbers respectively. Introducing these

    formulas in the Euler equation (eq. 3.45) yields the Kohn-Sham orbital equations:

    iiieffi r =

    + )(2

    1 2 (3.48)

    with:

    )(''

    )'()()( rrd

    rr

    rrr xceff

    +

    += (3.49)

    In which the first term is the external potential; the second term, the potential due to electron-electron

    repulsion and )(rxc , the exchange-correlation potential, given by:

    [ ]

    )(

    )(

    r

    rExcxc

    = (3.50)

    with [ ])(rExc the unknown exchange energy density functional.

    The resulting orbitals i are the Kohn-Sham orbitals and are used to construct the electron density. As

    can be seen from equations 3.48 and 3.49, the Kohn-Sham equations are nonlinear and have to be solved

    iteratively. Computationally, solving the Kohn-Sham equations is not much more demanding than

    solving the Hartree Fock equations. The Kohn-Sham theory, exact in principle, differs from the Hartree-

    Fock theory in its capacity to fully incorporate the exchange-correlation effect of the electrons. Note

    however that at this stage the exchange-correlation part remains unknown.

    Chapter III: Theoretical background

    3 4 The exchange-correlation energy functionals

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    46/199

    31

    3.4 The exchange correlation energy functionals3.4.1 Introduction

    An explicit form of [ ])(rExc is needed to solve the Kohn-Sham equations. The search for an accurate[ ])(rExc has encountered tremendous difficulties and continues to be a great challenge in Density

    Functional Theory. The difference between DFT methods is the choice of the functional form of the

    exchange-correlation energy.

    [ ])(rExc is generally separated into the exchangeEx and the correlationEc parts:

    [ ] [ ] [ ])()()( rErErE cxxc += (3.51)The correlation between electrons of parallel spin is different from this between electrons of opposite

    spin. The exchange energy is given by the sum of contributions of the and spin densities, as

    exchange involves only electrons of the same spin:

    [ ] [ ] [ ])()()( rErErE xxx += (3.52)

    [ ] [ ] [ ] [ ])(),()()()( rrErErErE cccc ++=

    (3.53)

    The total density is the sum of the andcontributions. The exchange-correlation functional can also

    be written as follows:

    [ ] [ ] [ ]( ) [ ] [ ]( ))()()()()( rJrErTrTrE eeSxc += (3.54)

    Herein, the first term is the contribution to the correlation energy of the kinetic energy obtained as the

    difference between the kinetic energy for the non-interacting system [ ])(rTS as calculated in the

    Kohn-Sham approximation and the exact kinetic energy for the interacting system [ ])(rT .

    The second term contains both a correlation and an exchange contribution to the exchange-correlation

    energy.

    3.4.2 Hybrid methods

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    47/199

    32

    y

    In this work, we use the Becke 3-Parameter (exchange), Lee, Yang and Parr (correlation) (B3LYP)22,23

    functional as exchange-correlation functional. This is a hybrid functional in which the exchange-

    correlation functional is divided in an exact exchange energy term and an exchange energy termfounded in a local density approach (LDA), but gradient corrected.

    3 880 0(1 ) (1 )

    B LYP

    XC

    LSDA HF B LYP VWN cX X X X C C CE a E a E a E a E a E = + + + +

    (3.55)

    In eq. 3.55,EXLSDAis the exchange energy obtained from the local spin density approximation, EC

    VWN is

    the standard local correlation functional obtained by Vosko, Wilk en Nusair, ECLYP is the gradient

    corrected functional for the correlation energy obtained by Lee, Yang and Parr,88B

    XE is a correction on

    the LSDA exchange energy and EXHF

    is the exact Hartree-Fock exchange energy. a0, ax and aC areempirical coefficients obtained by a least-square fit to experimental data.

    The B3LYP method is known to give good results for several physical observables and is less

    demanding than post-Hartree-Fock methods. Also, it is a widely used method allowing for direct

    comparison with other work.

    3.5 The chemical potentialThe physical significance of the Lagrange multiplier from the Euler equation (eq. 3.45) can be

    clarified by considering the total differential of the energy [ ])(, rNE for the change from one groundstate to the other:

    rdrdr

    EdN

    N

    EE

    Nr

    )()()(

    +

    = (3.56)

    This expression must be the same as the total differential ofEusing(r) and (r) as the basic variables:

    rdrdr

    Erdrd

    r

    EE

    rr

    )()(

    )()(

    )()(

    +

    = (3.57)

    Chapter III: Theoretical background

    The ground state(r) must satisfy the Euler equation (eq. 3.45), as such:

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    48/199

    33

    g ( ) y q ( q ),

    ==

    )()(

    rr

    Econstant (3.58)

    and:

    = dNrdrd )( (3.59)

    Inserting eq. 3.58 and eq. 3.59 in eq. 3.57 gives:

    rdrd

    r

    EdNE

    r

    )(

    )( )(

    += (3.60)

    Comparing eq. 3.56 and eq. 3.60, one obtains:

    )(rN

    E

    = (3.61)

    In analogy with the chemical potential in thermodynamics (replaceEby G andNby n at constantp and

    T), the Lagrange multiplier is called the electronic chemical potential.

    The electronic chemical potential measures the escaping tendency of an electron from the electronic

    cloud (cfr. the chemical potential in thermodynamics measuring the energy change when infinitesimal

    amounts of a given substance are added or withdrawn from the system under certain conditions). With

    the interpretation of the Lagrange multiplier in the Euler equation as the chemical potential, the

    conceptual DFT was found.

    Assuming a quadratic relationship between the energy Eand the number of electrons N (Fig. 3.1) the

    finite-difference approximation tofor a system can be written as:

    2

    EAIE+= (3.62)

    In whichIEandEA indicate the ionization energy and electron affinity respectively.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    49/199

    34

    Figure 3.1:E versusN plot for a typical chemical species.

    The expression for the chemical potential is the opposite of the expression proposed by Mulliken for the

    electronegativity:

    2

    EAIEM

    += (3.63)

    3.6 Chemical potential derivativesAfter the introduction of the chemical potential, other reactivity descriptors were identified. They

    quantify the response of the energy of a system on a perturbation in the number of electrons and/or

    chemical potential. Figure 3.2 gives an overview of all derivatives)(' rN

    E

    mm

    n

    up to the second order

    )2( n together with the identification or definition of the corresponding response function.

    slope=IE

    slope=EA

    N0 N0 +1N0 1

    N

    E

    IE

    EA

    Chapter III: Theoretical background

    [ ])(, rNE

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    50/199

    35

    [ ])(,

    ==

    )(rN

    E )()(

    rr

    EN

    =

    =

    =

    NN

    E

    r)(

    2

    2

    )()(

    )( )(

    2

    rfN

    r

    rN

    E

    r

    =

    =

    ( )',)'(

    )(

    )'()(

    2

    rrr

    r

    rr

    E

    NN

    =

    =

    Figure 3.2: Energy derivatives and response functions in the canonical ensemble.

    )(' rN

    Emm

    n

    )2( n

    The response of the chemical potential to an external perturbation can be expressed as the total

    derivative of [ ])(, rN :

    rdrdr

    dNN Nr

    )()()(

    +

    = (3.64)

    This is the basic equation for the definition of reactivity descriptors used by the interpretation of the

    reactivity of different reaction partners. In the next paragraphs we discuss the reactivity indices

    important in this work.

    3.6.1 Hardness and softness

    The first term of eq. 3.64 is the curvature of the plot in figure 3.1 and defines the global hardness:

    )(

    2

    2

    )( 2

    1

    2

    1

    rrN

    E

    N

    =

    = (3.65)

    The reciprocal ofis the global softness S24:

    1=S (3.66)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    51/199

    36

    2=S (3.66)

    The finite difference approximation forand Sis given by:

    2

    EAIE= (3.67)

    and:

    EAIES

    =

    1 (3.68)

    The expression for the global hardness (eq. 3.67) equals half of the reaction energy for adisproportionation reaction:

    M + MM+ + M (3.69)

    Consequently, the global hardness is the resistance of the chemical potential to changes in the number of

    electrons of the system. The finite-difference approximation for the global hardness is approximately

    equal to the band gap, which is the energy difference between the lowest unoccupied molecular orbital

    (LUMO) and the highest occupied molecular orbital (HOMO) in the frontier molecular orbital theory.

    When the gap is large (high ), the stability of the system is high and the reactivity is low and viceversa.

    Looking at the definition of the global softness S as the inverse of the global hardness, a local

    counterpart of this quantity can be introduced24:

    )()()(

    )()()(

    rrr

    N

    N

    rrrs

    =

    = (3.70)

    The local softness )(rs gives the distribution of the global softness of the system. )(rs integrates to the

    global softness S:

    S= s(r)dr (3.71)

    Chapter III: Theoretical background

    3.6.2 Fukui function

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    52/199

    37

    The second term of eq. 3.64, the derivative of the chemical potential with respect to the external

    potential )(r yields a local quantity (i. e. varying from point to point) )(rf , the Fukui function25:

    )(

    )(

    )()(

    rNN

    r

    rrf

    =

    = (3.72)

    The Fukui function can be viewed as the sensitivity of a system's chemical potential to an external

    potential perturbation at a particular point r. Alternatively, )(rf can be seen as the change of the

    electron density )(r at each point rwhen the total number of electrons N is changed at a constant

    external potential.As )(r is expected to be discontinuous with respect to the number of electronsN, the use of different

    reactivity descriptors was proposed for electrophilic and nucleophilic attacks.

    For a system ofNo electrons, the left derivative can be used when N increases from No to No + andmeasures the reactivity towards an electrophilic attack:

    =

    )(

    )()(

    rN

    rrf

    (3.73)

    the right derivative can be used whenNdecreases fromN0 toN0 - and measures the reactivity towardsa nucleophilic attack:

    ++

    =

    )(

    )()(

    rN

    rrf

    (3.74)

    In a finite difference approximation and for a system ofN0 electrons, these functions become:

    f(r) N0 N0 1

    (3.75)

    f+(r) N0 +1 N0

    (3.76)

    When integrating the Fukui function over atomic regions one finds the condensed Fukui functions for

    the nucleophilic and the electrophilic attack on atom A:

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    53/199

    38

    the nucleophilic and the electrophilic attack on atom A:

    fA = q (N0 ) qA (N0 1)

    (3.77)

    fA+ = q (N0 + 1) qA(N0 ) (3.78)

    qA(N0), qA(N0+1) and qA(N0-1) are the atomic populations for atom A in the neutral molecule

    (N0 electrons) and the corresponding anion (N0+ 1) or cation (N0- 1), all evaluated at the geometry of

    the neutral molecule or more generally at the geometry of the N0 electron system (cf. the demand for

    constant external potential in eq. 3.64).

    Combining eq. 3.64, eq. 3.70 and eq. 3.72, the local softness can be written as:

    Srfrs )()( = (3.79)

    As a direct consequence of eq. 3.77 and eq. 3.78 two types of local (condensed) softness are defined:

    s

    (r) = Sf(r) (3.80)

    s+

    (r) = Sf+(r) (3.81)

    3.6.3 Electrophilicity

    A quantitative measure of the electrophilicity of a species provides another useful tool for the

    rationalization of chemical reactivity. Starting from the question to what extent electron transfer

    contributes to the lowering of the total binding energy by a maximal influx of electrons, Parr et al.26

    provide validation for the qualitative suggestion made by Maynard et al.27 for the electrophilic power of

    a ligand.

    Based on a second order model for the change of the electronic energy Eas a function of the changesof the number of electrons N, at constant external potential (r), namely:

    2

    2NNE

    +=

    (3.82)

    with the electronic chemical potential and the chemical hardness, the electrophilicity index may

    be obtained by minimizing Ewith respect to N( 0=

    N

    E).

    Chapter III: Theoretical background

    The maximum electron-transfer equals:

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    54/199

    39

    maxN

    =

    (3.83)

    and the associated stabilization energy:

    2

    2E

    =

    (3.84)

    which is identified as the electrophilicity .

    In a finite-difference approximation, using a quadratic model for theEversusNplot, can be written

    as:

    )(8

    )(2

    EAIE

    EAIE

    +

    =

    (3.85)

    Eq. 3.85 indicates that depends on the electron affinity. However,EA quantifies the ability to accept

    exactly one electron, while is related to the maximum electron flow.

    depends on the hardness and the chemical potential, both global properties, making alsoa global

    quantity. A local counterpart can be identified based on the additivity of the global softness:

    +===k

    ksS22

    2

    2

    (3.86)

    The local electrophilicity is then given by:

    ++ = kk f (3.87)

    For recent extensions to the spin polarized case see ref. 28.The electrophilicity measures the reactivity towards an electrophilic attack. The nucleophilicity can be

    considered as the analogous reactivity descriptor for a nucleophilic reaction. However, a suitable

    expression for the nucleophilicity has not been identified yet.

    3.6.4 Nucleofugality

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    55/199

    40

    The nucleofugality29 represents the ability of a group of atoms (the nucleofuge) to act as a leaving

    group. It is related to the molecular fragments ability to accept an electron, since a nucleofuge takes an

    electron with it upon dissociation. Prior to its expulsion, the nucleofuge is covalently linked to theelectrophilic part of the molecule. Consider this part as a perfect electron donor, which transfers its

    electron without a barrier to an acceptor. The nucleofuge will accept an amount of charge (qideal) uponcontact with this perfect electron donor. Therefore, the charge on the nucleofuge equals q+qideal whencovalently bound to the electrophilic part of the molecule. Upon dissociation, a nucleofuge must take an

    entire electron with it, and thus changes its charge from q+qideal to q-1, leading to a destabilizationenergy nucleofugeE defined as the difference in energy between the product q-1 and the reactant

    q+qideal

    :)()1( idealnucleofuge qqEqEE +=

    (3.88)

    This equation can be rewritten as:

    2

    )( 2+=+= EAEnucleofuge

    (3.89)

    indicating that the destabilization energy nucleofugeE is related to the electron affinity.In a finite-difference approximation eq. 3.89 can be expressed in terms of the vertical IEandEA of the

    nucleofuge:

    )(8

    )3( 2

    EAIE

    EAIEEnucleofuge

    =

    (3.90)

    The nucleofugality is inversely related to the destabilization energynucleofuge

    E (eq. 3.89) - which is a

    kind of activation energy to overcome when a molecule is forced to accept an entire electron29:

    nucleofugeEe (3.91)

    with = 1.841 eV-1 (has been chosen so that the nucleofugality of the hydride anion is equal to 1, forfurther details, see ref. 29).

    Chapter III: Theoretical background

    The ability of a nucleofuge to act as a good or bad leaving group depends on the position of the

    minimum of theEversusNcurve. If qideal< -1 the nucleofuge is a perfect leaving group. If qideal> -1

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    56/199

    41

    minimum of the versus N curve. If q 1 the nucleofuge is a perfect leaving group. If q 1

    energy is needed to split off the nucleofuge.

    A descriptor related to the nucleofugality is the electrofugality29. This is the energy needed to withdraw

    an electron from a molecular fragment as compared to the case of a perfect electron donor:

    )()1( idealeelectrofug qqEqEE ++= (3.92)

    or in terms ofIEandEA:

    )(8

    )3( 2

    EAIE

    EAIEE eelectrofug

    =

    (3.93)

    The nucleofugality and the electrofugality can be used to assess the thermodynamic stability of the

    electrofuge and nucleofuge. The nucleofugality indicates the relative stability of an electronacceptorNq-1

    compared to the acceptorfragment idealqq

    N+

    in the presence of a perfect electrondonor. Analogous, the

    electrofugality indicates the relative stability of an electrondonorEq+1 compared to the donorfragmentidealqqE

    +.

    The electrophilicity, electrofugality and nucleofugality form a complete set of reactivity indices. They

    quantify the relative energy of respectively the reference system N0, the corresponding cationN0+1 and

    anionN0-1 in contact with a perfect electrondonor. Figure 3.3 gives an overview.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    57/199

    42

    E E N E N +N electrofuge 0 0 ideal = ( -1) - ( )

    = ( ) - ( )E N E N +N0 0 ideal

    IE E N E N = ( -1) - ( )0 0

    E E N E N +N nucleofuge 0 0 ideal = ( +1) - ( )

    EA E N E N += ( ) - ( 1)0 0

    E

    N0-1 N0 N0+1

    Figure 3.3: E versusN plot indicating the relation between the ionisation energy (IE), electron affinity (EA),

    electrophilicity (), nucleofugality ( nucleofugeE ) and electrofugality ( eelectrofugE ).

    3.7 Hard and soft acids and bases (HSAB) principleThe concepts of hardness and softness of a system were already introduced by Pearson in the 1960s30.

    They were at that time used in the explanation of acid-base reactions in their most general form (Lewis

    acid/Lewis base).

    Pearson stated that:Hard acids prefer to react with hard bases whereas soft acids prefer to interact with

    soft bases. This is known as the hard and soft acids and bases (HSAB) principle.

    According to this principle and in analogy with earlier work by Gzquez 31 and our group32 and its

    generalization by Ponti33, the (preferred) reactivity between the reaction partners can be based on the

    difference in local softness s(r) of the interacting parts (atoms, functional groups,...) of these reaction

    partners:

    s(r) = |s+(r) s-(r)| (3.94)

    which should be minimal for optimal interaction, a criterion used throughout this work.

    Chapter III: Theoretical background

    To quantitatively predict reaction rates, one should locate the transition states and compute activation

    energies, which is a difficult task. The HSAB principle offers the advantage that the characteristics of a

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    58/199

    43

    reaction (mainly kinetic aspects) are described in terms of the properties of the reagents in the ground

    state, without explicit numerical calculation of characteristics along the reaction path. The use of the

    HSAB principle is indeed based on a perturbational ansatz as formulated by Parr7

    . Assuming thatreaction paths of similar reactions (eg. differing only in substitution pattern of one reagent) wont cross

    (Klopmans rule34), the relative energies at the beginning of the reaction can be expected to predict a

    sequence of activation energies. As such, application of the HSAB principle allows the deduction of

    relative activation energies from information on the reactant properties only. This principle offers the

    possibility to interpret and to predict the results of reaction path calculations going along with Parrs

    dictum: To compute is not to understand35.

    4. Basis sets

    4.1 Slater and Gaussian type orbitalsAs discussed in previous sections, the molecular orbitals in the Hartree-Fock treatment are developed as

    a linear combination of nuclear-centred basis functions. There are two types of basis functions used for

    electronic structure calculations. The first type being the Slater Type Orbitals (STO)36:

    nlmST (r,,;) =Nnorrn lerYlm (,) (3.95)

    in which Nnor is a normalisation constant, n, l and m are the quantum numbers, the Slater orbital

    exponent, r, and the spherical coordinates of the electron relative to the nucleus, and the functions

    Ylm(,) are the spherical harmonics. The exponential dependence on the distance between the nucleus

    and the electron mirrors the exact orbital for the hydrogen atom. However the calculation of three and

    four-centred electron integrals cannot be performed analytically37.

    The second type of orbitals that is mostly used are the Gaussian Type Orbitals (GTO)38

    :

    GT

    (x,y,z;) = Nnorxpy

    qz

    se

    r2

    (3.96)

    withNnora normalisation constant,p, q ands positive integers (p + q + s = l)and the Gaussian orbital

    exponent.

    As can be seen from eq. 3.95 and eq. 3.96, these two types of basis functions show a different radial

    behaviour. The GTO behaviour near the nucleus and at long distances is however incorrect: the GTO

    falls off too rapidly far from the nucleus and the r2dependence in the exponential makes the GTOpoor

    to represent the proper behavior near the nucleus. The STO shows the correct radial behaviour, but the

    l i l l i i h b i f i b l diffi l k f

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    59/199

    44

    two-electron integral evaluation using these basis functions becomes an extremely difficult task for

    polyatomic molecules. GTOs on the other hand are easier to evaluate as e.g. the product of two GTOs

    result in one GTO. In practice, a compromise is sought between the computational efficiency of theGTO's and the correct form of the STO's. Therefore, a number of Gaussians (called uncontracted or

    primitive functions) is contracted in a linear combination to fit a STO39:

    CGT = Cjj (x,y,z;j )

    j=1

    K

    (3.97)

    withKthe degree of contraction.

    The expansion coefficients and orbital exponents of the primitive Gaussians in eq. 3.97 can now be

    optimized so thatCGT

    approximates a Slater type function, leading to an STO-KG basis set.

    4.2 Minimal basis setsThe use of the minimal basis sets (i.e. a basis set containing just enough basis functions to accomodate

    all the electrons of the atom) constitutes the simplest level of ab initio molecular orbital theory40-42. The

    essential idea of the minimal basis set is selecting one basis function for every atomic orbital includingall sub shells. For hydrogen, the minimum basis set is just one 1s orbital. For carbon, the minimum basis

    set consists of a 1s orbital, a 2s orbital and a set of three 2p orbitals. The most common minimum basis

    sets is the STO-nG type, in which n primitive GTOsare combined to fit to a STO orbital.

    The next step up in basis set size is Triple Zeta (TZ), which has three times the number of functions as

    the minimal basis set, Quadruple Zeta (QZ), Quintuple Zeta (5Z)

    4.3 Split valence basis set

    As seen in the previous paragraph, minimal basis sets can be built from a combination of n primitive

    GTOs. In the same way, Pople et al.42-47 designed the split valence basis set. They are denoted as k-

    nlmG basis sets in which each core orbital is represented by a single contraction of kprimitive GTOs

    while the valence orbital is split into three regions, represented by n, l and m primitive GTOs

    respectively42-47.

    Chapter III: Theoretical background

    4.4 Polarization functions, diffuse functions

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    60/199

    45

    A better representation of the chemical bonding can be obtained by adding polarization functions. For

    example, the chemical bond involving a hydrogen atom is poorly described by using only its s-orbital,

    because of the spherical symmetry of thes-functions. The electron distribution along the bond should beclearly different than in any other direction. Consequently, adding a contribution of a p-orbital to thes-

    orbital will improve the description of the bond. For the same reason d-type functions are needed for the

    first row atoms. The presence of polarization functions is denoted by the symbol: '*', which means that

    polarization functions are only added top-functions. ** denotes that polarization functions are added

    to s- and p-functions. If methods including electron correlation are used, polarization functions are

    needed. Indeed, to describe the situation in which two electrons are on opposite sides of the nucleus, one

    needs functions with the same magnitude of exponents, but with different angular momenta.

    For the same reason, one can add diffuse functions, i.e. functions with small exponents. The primary

    argument is to extend the valence region of an atom, which is very convenient when considering

    systems such as anions or exited states. Diffuse functions are denoted by + or ++. The first +

    indicates that one set of diffuses andp function is added to heavy atoms, the second + indicates that a

    diffuses-function is added to hydrogen atoms.

    5. Molecular quantities

    5.1 The electron density functionThe electron density function )(r is defined as the probability to find an electron in the volume dr

    around r. In terms of the wave function, this function can be expressed as:

    (r) = N ... * (x1,x2,...,xN) (x1,x2 ,...,xN)d1dx 2,...,d xN (3.98)

    in which the integration is performed over all the variables of the wave function, except for the spatial

    coordinates of one electron.

    This function must integrate to the total number of electronsNwithin the molecule:

    (r)dr=N (3.99)

    Within the Hartree-Fock approximation (RHF case), eq. 3.99 yields:

    ( ) *

    K

    K

    (3 100)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    61/199

    46

    (r) = P=1

    =1

    (3.100)

    in which the summation is done over and v of all b asis functions.

    5.2 The atomic electron populationChemists often like to ascribe portions of the total electronic charge to specific atoms in molecules.

    There are several ways of achieving this. In the following paragraphs the population analysis methods

    important for this study will be discussed.

    5.2.1 Orbital-based population analysis methods: the natural populationanalysis method

    The natural population analysis method (NPA), developed by Reed, Weinstock and Weinhold48,

    attempts to define atomic orbitals based on the molecular wave function. As a result, atomic orbitals are

    obtained depending on the chemical environment of the atom. This approach is based on the first order

    reduced density matrix )',( 11 xx , defined as:

    NNN xdxddxxxxxxxx ...),...,,(),...,,(...),( 212'121

    *'11 = (3.101)

    The orbitals resulting from the diagonal reduced density matrix are called the natural orbitals and the

    diagonal elements are the occupation numbers. These natural orbitals are orthonormal molecular orbitals

    having maximum occupancy. The natural atomic orbitals are, by analogy, the atomic orbitals having

    maximum occupancy and are obtained as eigenfunctions of the atomic subblocks of the density matrix.

    Reed, Weinstock and Weinhold now defined these subblocks and obtained eigenfunctions that are

    orthonormal, not only within the subblock, but also with all the other eigenfunctions leading to the NPAcharges.

    5.2.2 Electrostatic potential derived charges

    An alternative method for obtaining atomic charges is to fit the electrostatic potential to a series of point

    charges centered on the atomic nuclei. This monopole expansion VMis given by:

    Chapter III: Theoretical background

    VM(r) =qk

    rRkk

    (3.102)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    62/199

    47

    The best least squares fit is obtained by minimizingy :

    2

    21 ))()((),...,,( iMm

    i

    ik rVrVqqqy = (3.103)

    with the constraint that the total molecular charge should be preserved. The electrostatic potential

    derived charges used in this work were obtained by the so-called ChelpG method, designed by

    Breneman and Wiberg49.

    6. Solvent effects

    Solvent effects play an important role in determining equilibrium constants, selectivity and

    conformational behaviour. The large majority of methods describing chemical processes in solution are

    based on continuum models involving a bulk dielectric constant for the solvent and a cavity surrounding

    the solute molecule50. The shape and size of the cavity are differently defined in the various versions of

    the continuum models. The optimal size and shape of the cavity have been subject of debate, and several

    definitions have been proposed.

    The cavity should exclude the solvent and should contain within its boundaries the largest possible part

    of the solute charge distributionM. Obviously these requirements are in contrast with the description of

    the whole system given by any quantum chemical level. The electronic charge distribution of an isolated

    molecule, in fact, persists to infinity. In a condensed medium the conditions onMat large distances are

    less well-defined, but in any case there will be an overlap with the charge distribution of the medium,

    not explicitly described in continuum models but existing in real systems.

    It is universally accepted that the cavity shape should reproduce as well as possible the molecular shape.

    Cavities not respecting this condition may lead to deformations in the charge distribution after solvent

    polarization, giving large unrealistic effects on the results, especially on properties. Here, once again,

    there is a trade-off between computational exigencies and the desire for better accuracy. Computations

    are far simpler and faster when simple shapes are used, such as spheres and ellipsoids, but molecules are

    often far from having a spherical or ellipsoidal shape. In the following paragraphs, the widely used

    polarizable continuum model and an improvement of this model are discussed.

    6.1 The PCM model51 53

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    63/199

    48

    In the polarizable continuum model (PCM)51-53, the solute cavity is defined by a set of overlapping

    spherical atoms having the Van der Waals radius multiplied with a constant, since the first hydratation

    shell has dielectric properties different of those of bulk solvent. The surrounding solvent is representedby an infinite, unstructured, polarizable dielectric medium outside the boundaries of the cavity.

    The general basis of the PCM model can be expressed in the following way: the Hamiltonian of the

    system is partitioned into two parts, regarding solute (M) and solvent (S), supplemented by a coupling

    term:

    MSSMtot HHHH ++= (3.104)

    In practice, HS is discarded and the attention is put on the Hamiltonian regarding M, with inclusion ofthe M-Scoupling as an effective interaction operatorVint:

    int

    0 VHH MM += (3.105)

    Vint results from the charge distribution in the cavity M that polarizes the continuum, which in turn

    polarizes the solute charge distribution. It is the sum of the electrostatic potential VM generated by the

    charge distribution M in the cavity and the reaction potential V generated by the polarization of the

    dielectric medium.

    MVVV +=int (3.106)

    To obtain the reaction potential V, the cavity surface is approximated in terms of a set of finite elements

    (called tesserae) small enough to consider the apparent surface charge (r) almost constant within each

    tessera. With (r) completely defined point-by-point, it is possible to define a set of point charges qkon

    each of these tesserae in terms of the local value of(r) multiplied by the corresponding areaAk.

    = k kk

    k

    k

    rrq

    rrArV )( (3.107)

    qk depends on the dielectric constant that characterizes the solvent and on the electric potential Vint

    describing the solute-solvent interactions. Since neither Vint nor qk are known initially, one finds the

    apparent surface charges by an iterative process. The converged charges are used to find the potential

    energy of electrostatic interaction.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    64/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    65/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    66/199

    CHAPTER IVA computational and conceptual DFT study

    on the Michaelis complex of pI258 arsenate reductase:structural aspects and activation

    of the electrophile and nucleophile

    The human mind treats a new idea the way the body treats a strange protein it rejects it.

    (Peter Medawar)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    67/199

    52

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    68/199

    (Fig. 4.1) is observed in all the X-ray structures of pI258 ArsC present in the PDB1,11. ArsC also possesses an -helix (extending from amino acid 16 to 29)1 of which the N-terminal side faces thenucleophile. Since the experimental determination of the pKa of the Cys10 thiol group is not

    i h f d ( Ch II) hi h l l h i l l l i ill l h ff f h

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    69/199

    54

    straightforward (see Chapter II), high level quantum chemical calculations will reveal the effect of the

    C10-K+

    interaction network and the macro-dipole arising from the -helix on the pKa of the Cys10functional group.

    Figure 4.1: C10-K+ interaction network. WT model. The Ser17 mutant (-Ser17), the Asn13 mutant(-Asn13), the potassium mutant (-K+) and the double mutant (-Asn13/- K+) are constructed from this model.In all of the model systems, the hydrogen atoms were optimized at the B3LYP/6-31+G* level. Thecoordinates of the heavy atoms are taken from the PDB structure 1JF8 ref. 1 (See also 2.2: Interactions withthe nucleophile).

    All essential intermediates in the reaction mechanism of ArsC have been visualized with X-raycrystallography supplemented by NMR12, with the exception of a Michaelis complex. We will focus on

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    70/199

    were partitioned into two layers (Fig. 4.2). The most relevant parts, being the nucleophile and thesubstrate, form the inner layer and and were treated at a high level of theory (B3LYP/6-31+G**) whilethe remaining part of the system, the ligand binding loop, constituting the outer layer was described by acomputationally less demanding method (HF/6 31G)

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    71/199

    56

    computationally less demanding method (HF/6-31G).

    Figure 4.2: Reduction of the X-ray structure of ArsC (PDB 1LJU)12 to the wild type (WT) model.Partitioning of the WT model system of ArsC into 2 layers: high level represented in Ball & Stick; low levelin Tube. A similar division is made for the Asn13Ala and the Arg16Ala mutants.

    2.2 Interactions with the electrophileFor comparison of the charge distribution between enzyme bound and free arsenate in the gas phase, the

    NPA population analysis21 calculated at the B3LYP/6-31+G** level, is used. This choice is founded onthe many successful applications of this population analysis in the study of molecular properties22. In

    contrast, Mulliken charges are not advisable to use because of their strong basis set dependence23,whereas electrostatic potential derived charges (e.g. ChelpG) are not recommended in view of theChelpG charge-deriving scheme24 from which one can suspect a poor description of ligands embeddedin large systems25, our point of interest.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    72/199

    To explore and quantify the effect of the C10-K+ interaction network and the -helix on the basicity ofCys10,we calculated the proton affinity of Cys10 in the presence and absence of the components of theC10-K+ interaction network and the -helix. Proton affinities were calculated at the B3LYP/6-31+G**level by subtracting the energies of the optimized (B3LYP/6-31+G*) protonated and deprotonated

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    73/199

    58

    level by subtracting the energies of the optimized (B3LYP/6 31+G ) protonated and deprotonated

    forms. To translate changes in proton affinity to changes in the acid dissociation constant (pKa), wecalculated proton affinities of a series of five thiolates (methanethiol, benzenemethanethiol,mercaptoethanol, cysteine and trifluoroethanethiol) and plotted these values against experimental pKavalues (Fig. 4.3). The resulting linear relationship was used to extrapolate the pKa of Cys10 in theconsidered models from its calculated proton affinity.

    R=0,882

    5

    6

    7

    8

    9

    10

    11

    -360 -355 -350 -345 -340 -335

    p

    a

    K

    Proton affinity (kcal/mol)

    1

    2

    3

    4

    5

    6,2

    10,3

    Figure 4.3: Proton affinity-pKa correlation curve calculated in the gas phase for a series of fivesubstituted thiolates.

    1 = methanethiol; 2 = benzenemethanethiol; 3 = mercaptoethanol; 4 = cysteine; 5= trifluoroethanethiol.Effect of the hydrogen bond network on the pKa of Cys10 is shown. The calculated proton affinities ofCys10 in the presence of different elements of the C10-K+ interaction network are inserted.

    10.3 pKa of methanethiol10.0 pKa of methanethiol in the presence of a solitary hydrogen bond with Ser176.2 pKa of methanethiol in the presence of the C10-K+ interaction network6.0 pKa of methanethiol in the presence of the C10-K+ interaction network + -helix dipole effect

    Chapter IV: Michaelis complex of pI258 ArsC

    2.4 DFT Reactivity analysisIsolated structures of H3AsO4/H2AsO4

    -/HAsO42-/AsO4

    3- and CH3S- were optimized in gas phase and

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    74/199

    59

    Isolated structures of H3AsO4/H2AsO4 /HAsO4 /AsO4 and CH3S were optimized in gas phase and

    solution (SCI-PCM model)30

    with = 20.7 on the B3LYP/6-31+G** level. Thiolate and arsenate aresoft (polarizable) species. As a consequence a soft-soft mechanism underlies the reactivity of ArsC,which at the local level can be quantified by the difference in local softness (HSAB principle) 31 of theinteracting parts s(r) = |s+(As) s-(S)| (eq. 3.94, Chapter III).

    An electrostatic model was used as an approximation for the influence of ArsC on the reactivity indicesof arsenate and CH3S

    - (as model for Cys10). Arsenate was embedded in the enzymatic environment ofthe WT and Arg16Ala model systems, while the environments of WT and Asn13/-K+ were used tosurround CH3S

    -. The enzymatic environment of wild type and mutant ArsC was represented by ChelpGpoint charges, calculated at the B3LYP/6-31G** level24,29.

    NPA charges21 calculated at B3LYP/6-31+G** were used to obtain the Fukui function. The global

    softness is calculated usingHOMOLUMO

    S

    =1 (ref. 32) and the local electrophilicity:

    )(8

    )( 2

    EAIE

    fEAIE A

    +=

    ++

    (eq. 3.85 and 3.89, Chapter III).

    All calculations were performed using the GAUSSIAN 03 package33.

    3. Results and Discussion

    3.1 Theoretically optimized Michaelis complex3.1.1 Calculated model

    Starting from the X-ray structure of ArsC complexed with arsenite i. e. the product of the first reactionstep (PDB 1LJU)12 (Fig. 2.3, Chapter II), a model of the enzyme-substrate (Michaelis) complex of ArsCis optimized using a 2-layer QM/QM ONIOM17-20 scheme (B3LYP/6-31+G**//HF/6-31G) (Fig. 4.2).To check whether this structure is an acceptable Michaelis complex, the same methodology used toobtain the ArsC Michaelis complex was also applied to a product-like structure of a protein tyrosine

    phosphatase (PTPase) of the Yersiniabacteria in complex with NO3- (PDB 1YTN)34, which is analogous

    to AsO3-. This calculated PTPase Michaelis complex was compared with the experimental X-ray

    structure of a complex with a tetrahedral oxyanion (Michaelis complex-like structure) of the same

    Yersinia PTPase (PDB 1YTS)35. All the observed enzyme-substrate interactions in 1YTS were retrievedin the optimized Michaelis complex. In analogy with this result, the structure of the ArsC Michaeliscomplex obtained from 1LJU by using the QM/QM ONIOM scheme (B3LYP/6-31+G**//HF/6-31G)can be treated with confidence.

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    75/199

    60

    The dihedral angles of the peptide bonds found in the optimized enzyme-substrate complex of ArsCdeviate on average by seven degrees from planarity (between a deviation-maximum and -minimum ofeleven and two degrees respectively). Experimental statistical data report deviations from the exact

    planar peptide bond up to six degrees36a,b and even more when circular peptides are considered36c. Assuch, in the ligand binding pocket of ArsC, which has a circular geometry, the averaged deviation from

    peptide bound planarity can be considered as acceptable and the activation barriers of peptide bondrotations are properly described by the proposed ONIOM scheme.

    3.1.2 Enzyme-substrate interactions

    During ligand binding, the desolvation energy has to be overcome and entropy is lost by the stabilizationof the ligand-binding loop. To deal with this energetically costly process several favorable enzyme-substrate interactions are formed in the Michaelis complex of which figure 4.4 and table 4.1 present anoverview. All comparisons were performed with the X-ray structure of pI258 ArsC (PDB 1LJU)12,

    because this structure resembles most the Michaelis complex studied in this work.

    Donor--Acceptor l() a ()

    NH(Arg16)--OH(LG) 2.82 168OH(LG)--OH2(1) 2.70 147N16H--O(1) 2.86 153N17H--O(1) 2.79 166HOH(2)--O(2) 2.69 170N14H--O(2) 2.83 153N11H--O(3) 2.84 150NH(Arg16)--O(3) 2.74 164HOH(3)--O(3) 2.70 167

    Table 4.1: Enzyme-substrate interactions in the Michaelis complex. HOH(x)--O(y) points to ahydrogen bond between HOH number x as proton donor and the substrate oxygen atom numberyasproton acceptor. lgives the distance between donor and acceptor in ngstrm and a gives the anglebetween donor-proton-acceptor in degrees. LG stands for leaving group and NxH for backbone amidegroup of the amino acid with number x.

    Chapter IV: Michaelis complex of pI258 ArsC

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    76/199

    61

    Figure 4.4: Stereoview of the optimized (2-layer ONIOM scheme: B3LYP/6-31+G**//HF/6-31G)Michaelis complex. Figure created by Messens, J.

    To discern which anionic species of arsenate is most likely to be bound in the active site, we comparedthe interaction energies (calculated at B3LYP/6-31+G**) of mono- and di-anionic arsenate with ArsC.The binding of di-anionic arsenate turned out to be 82 kcal/mol more favourable than that of mono-anionic arsenate, despite the vicinity of the negatively charged Cys10.

    In the ArsC-di-anionic arsenate complex, all backbone amide hydrogen atoms in the catalytic loop areoriented toward the centre of the loop. With the exception of Gly12 and Asn13, they all form hydrogen bonds with the oxygen atoms of di-anionic arsenate. All free electron pairs of these oxygens areinvolved in hydrogen bonding. In the case of a mono-anionic arsenate, the extra hydrogen atom on oneof the oxygens would experience steric hindrance, making a di-anionic form of arsenate more favorable.The nucleophilic SCys10 interacts with the Gly12 and Asn13 backbone amide groups and with thehydroxyl group of Ser17 via hydrogen bonds (Fig. 4.4). In the crystal structure of the product of the firstreaction step (PDB 1LJU)12, the distance between SCys10 and the Gly12 amide, between SCys10 and

    the Asn13 amide, and between SCys10 and OSer17 are respectively 4.26 , 3.65 and 3.28 . Withthe exception of the first interaction, these distances are in the same range as those in the in silicoobtained Michaelis complex (Table 4.1).

    3.1.3 Arg16 guanidinium group

    In the optimized wild type model, the guanidinium group of Arg16 provides an extension of thesubstrate-binding pocket, via a spherical structure surrounding the substrate. In LMW PTPase9, a crucial

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    77/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    78/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    79/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    80/199

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    81/199

    Chapter IV: Michaelis complex of pI258 ArsC

    3.4 Activation of the nucleophileAt the optimum pH for enzymatic catalysis by ArsC (pH = 8.0)2, a substantial amount of free cysteine(pKa = 8.3) is present in the thiolate form. In the enzyme-substrate complex, however, the presence of

  • 8/3/2019 Goedele Roos- Theory meets experiment: a combined quantum chemical-experimental study of the reaction mechanism of pI258 arsenate reductase

    82/199

    67

    di-anionic arsenate in the vicinity of Cys10 is expected to increase the latters basicity and to drive thethiol/thiolate equilibrium toward the thiol form, which is a weaker nucleophile compared to the thiolateform40. However, it can be anticipated that the enzymatic environment favors the deprotonated state.Apart from the backbone amide hydrogens of Gly12 and Asn13, Ser17 i