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Title: Post-QM/MM Methodologies for Structural Refinement of Photosystem II and Other Biological Macromolecules

Article Type: Special Issue (GOVINDJEE)

Keywords: Quantum Mechanics, Molecular Mechanics, Photosystem II, EXAFS, MOD-QM/MM

Corresponding Author: Victor S. Batista, Ph.D.

Corresponding Author's Institution: Yale University

First Author: Eduardo M Sproviero, Ph.D.

Order of Authors: Eduardo M Sproviero, Ph.D.; Michael B Newcomer, PhD Student; Jose A Gascon, Ph.D.; Enrique R Batista, Ph.D.; Gary W Brudvig, Ph.D.; Victor S Batista, Ph.D.

Abstract: Quantum mechanics/molecular mechanics (QM/MM) hybrid methods are currently the most powerful computational tools for studies of structure/function relations and structural refinement of macrobiomolecules (e.g., proteins and nucleic acids). These methods are highly efficient since they implement quantum chemistry techniques for modeling only the small part of the system (QM layer) that undergoes chemical modifications, charge transfer, etc., under the influence of the surrounding environment. The rest of the system (MM layer) is described in terms of molecular mechanics force fields, assuming that its influence on the QM layer can be roughly decomposed in terms of electrostatic interactions and steric hindrance. Common limitations of QM/MM methods include inaccuracies in the MM force fields when polarization effects are not explicitly considered, and the approximate treatment of electrostatic interactions at the boundaries between QM and MM layers. This paper reviews recent advances in the development of computational protocols that allow for rigorous modeling of electrostatic interactions in extended systems beyond these common limitations of QM/MM hybrid methods. Emphasis is placed on the Moving-Domain QM/MM (MOD-QM/MM) methodology where the system is partitioned into many molecular domains and the electrostatic and structural properties of the whole system are obtained from an iterative self-consistent treatment of the constituent molecular fragments. We illustrate the MOD-QM/MM method as applied to the description of macromolecules of biological interest, including photosystem II and guanine DNA quadruplexes as well as in conjunction with the application of post-QM/MM optimization methods for direct comparisons with high-resolution spectroscopic data (extended X-ray absorption fine structure spectra, infrared spectra, exchange coupling constants and calculations of nuclear magnetic resonance chemical-shifts).

1

Post-QM/MM Methodologies for Structural

Refinement of Photosystem II and Other

Biological Macromolecules

Eduardo M. Sproviero · Michael B. Newcomer · José A. Gascón1 · Enrique R.

Batista+ · Gary W. Brudvig · Victor S. Batista*

Yale University, Department of Chemistry, P. O. Box 208107, New Haven Connecticut 06520-8107 USA; and +Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545.

e-mail: victor.batista@yale.edu

Abstract: Quantum mechanics/molecular mechanics (QM/MM) hybrid methods are currently the

most powerful computational tools for studies of structure/function relations and structural

refinement of macrobiomolecules (e.g., proteins and nucleic acids). These methods are highly

efficient since they implement quantum chemistry techniques for modeling only the small part of

the system (QM layer) that undergoes chemical modifications, charge transfer, etc., under the

influence of the surrounding environment. The rest of the system (MM layer) is described in terms

of molecular mechanics force fields, assuming that its influence on the QM layer can be roughly

decomposed in terms of electrostatic interactions and steric hindrance. Common limitations of

QM/MM methods include inaccuracies in the MM force fields when polarization effects are not

explicitly considered, and the approximate treatment of electrostatic interactions at the boundaries

between QM and MM layers. This paper reviews recent advances in the development of

computational protocols that allow for rigorous modeling of electrostatic interactions in extended

systems beyond these common limitations of QM/MM hybrid methods. Emphasis is placed on the

Moving-Domain QM/MM (MOD-QM/MM) methodology where the system is partitioned into

many molecular domains and the electrostatic and structural properties of the whole system are

obtained from an iterative self-consistent treatment of the constituent molecular fragments. We

illustrate the MOD-QM/MM method as applied to the description of macromolecules of biological

interest, including photosystem II and guanine DNA quadruplexes as well as in conjunction with

the application of post-QM/MM optimization methods for direct comparisons with high-resolution

spectroscopic data (extended X-ray absorption fine structure spectra, infrared spectra, exchange

coupling constants and calculations of nuclear magnetic resonance chemical-shifts).

Keywords Quantum Mechanics, Molecular Mechanics, Photosystem II, EXAFS,

MOD-QM/MM

1Current address: Department of Chemistry, University of Connecticut, Unit 3060, Storrs,

CT 06269.

ManuscriptClick here to download Manuscript: Brudvig_Batista.doc Click here to view linked References

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Abbreviations

BS Broken Symmetry

DFT Density Functional Theory

DNA Deoxyribonucleic acid

EE Electronic Embedding

ESP Electro Static Potential

EXAFS Extended X-Ray Absorption Fine Structure

FTIR Fourier Transform Infrared Spectroscopy

MEP Minimum Energy Path

MOD-QM/MM Moving-Domain-QM/MM

OEC Oxygen-Evolving Complex

P-EXAFS Polarized Extended X-Ray Absorption Fine Structure

PSII Photosystem II

QM/MM Quantum Mechanics-Molecular Mechanics

Introduction

In recent years, the development of computer hardware has allowed computational

chemistry methods to establish themselves as powerful tools for studies of

structure/function relations and structural refinement of macrobiomolecules (e.g.,

proteins and nucleic acids). In particular, quantum mechanics/molecular

mechanics (QM/MM) hybrid methods have become the most popular approaches

for modeling prosthetic groups that undergo significant structural or electronic

modifications when embedded in macromolecular structures (Warshel and Levitt

1976; Singh and Kollman 1986; Field et al. 1990; Maseras and Morokuma 1995;

Bakowies and Thiel 1996; Humbel et al. 1996; Svensson et al. 1996; Dapprich et

al. 1999; Philipp and Friesner 1999; Murphy et al. 2000b; Vreven and Morokuma

2000a; Vreven and Morokuma 2000b). The main aspect, common to all QM/MM

methods, is the partitioning of the system into two fragments: a small part (QM

layer) that includes the prosthetic group as described by quantum chemistry

methods and the rest of the system (MM layer) that is modeled in terms of

molecular mechanics force fields, since its influence can be roughly decomposed

in terms of steric hindrance and electrostatic interactions. This basic idea is at the

3

heart of the Moving-Domain QM/MM (MOD-QM/MM) methodology and the

applications studies reviewed in this paper (Gascon et al. 2006a; Menikarachchi

and Gascon 2008; Newcomer et al. 2009).

The various different versions of QM/MM methods differ in the way the

energy is evaluated, including additive and subtractive schemes; in the way the

boundary between QM and MM layers is treated, including link-atoms, localized

bond orbitals and the pseudopotential approach; on the specific level of quantum

chemistry and MM force field; and also on how the electrostatic interactions

between QM and MM layers are described, including mechanical embedding,

electronic embedding, and polarized embedding schemes. For example, in the

mechanical embedding approach, in vacuo QM calculations are coupled

classically to the MM layer via point charges at QM nuclear sites. In electrostatic

embedding, MM atoms polarize the QM layer. Finally, in the polarized

embedding scheme, the MM layer is polarized coupled to the QM charge density

(Gao and Xia 1992; Devries et al. 1995; Thompson 1996; Eichler et al. 1997;

Vreven et al. 2001) in the same spirit as in the MOD-QM/MM scheme (Gascon et

al. 2006a; Menikarachchi and Gascon 2008; Newcomer et al. 2009).

The MOD-QM/MM methodology partitions the system into many

molecular domains and obtains the electrostatic and structural properties of the

whole system from an iterative self-consistent treatment of the constituent

molecular fragments (Gascon et al. 2006a; Menikarachchi and Gascon 2008;

Newcomer et al. 2009). This involves a polarization scheme rendering a

description of prosthetic groups at the polarized embedding QM/MM level. Self-

consistent optimized structures and electrostatic-Potential (ESP) atomic charges

are obtained, modeling each fragment as a QM layer polarized by the distribution

of atomic charges in the surrounding molecular fragments. The resulting

computational task scales linearly with the size of the system, bypassing the

enormous demands of memory and computational resources that would otherwise

be required by ‘brute-force’ full QM calculations of the complete system.

Computations for several benchmark systems that allow for full QM calculations

have shown that the MOD-QM/MM method produces ab initio quality

electrostatic potentials and optimized structures in quantitative agreement with

full QM results (Gascon et al. 2006a; Menikarachchi and Gascon 2008;

Newcomer et al. 2009). The work reviewed in this paper illustrates the MOD-

4

QM/MM methodology as applied to the description of macromolecules of

biological interest, including photosystem II and guanine DNA quadruplexes, as

well as in conjunction with post-QM/MM structural refinement schemes based on

simulations of X-ray absorption spectra and direct comparisons with experimental

data.

The MOD-QM/MM method can be compared to a variety of other

alternative approaches, previously proposed for including polarization effects in

the MM layer of QM/MM calculations. Warshel and Levitt used a standard

atomic dipole description (Warshel 1976), also employed by others (Thompson

and Schenter 1995; Thompson 1996; Houjou et al. 2001; Illingworth et al. 2006)

where a dipole moment is induced in each atom of the MM layer, defined as the

product of the atomic polarizability and the electric field at the position of the

atom. Atomic polarizabilities are, thus, input parameters of the model. An ab

initio formulation of this approach is the so-called ‘effective fragment potential’

method (Chen and Gordon 1996; Day et al. 1996; Gordon et al. 2001), included in

the GAMESS quantum chemistry package (Schmidt et al. 1993). In this method,

the input parameters are charges, polarizabilities and higher moments previously

obtained from preliminary ab initio calculations, and the polarization of the

fragments is described by a standard distributed multipolar analysis. In contrast to

these approaches, the MOD-QM/MM does not require input parameters and

obtains ab initio quality electrostatic potentials based on self-consistent ESP

charges of the molecular domains, obtained by fitting their electrostatic potential

to the DFT QM/MM electrostatic potential computed over an extended set of grid

points around the QM layer.

Several schemes have been proposed for an explicit description of

polarization effects at the MM level, including several polarizable force fields

(Corongiu 1992; Bernardo et al. 1994; Banks et al. 1999; Stern et al. 1999;

Halgren and Damm 2001; Stern et al. 2001; Burnham and Xantheas 2002; Palmo

et al. 2003; Ren and Ponder 2003; Yu et al. 2003; Maple et al. 2005). However,

polarizable force fields remain computationally costly and their parametrization is

rather tedious. Among the various methods available for the description of

polarization effects, one approach is the ‘fluctuating charge model’ based on the

equalization of atomic electronegativities, studied by Rappe and Goddard (Rappe

and Goddard 1991), Berne (Rick et al. 1994; New and Berne 1995; Rick et al.

5

1995; Bader and Berne 1996; Rick and Berne 1996; Stuart and Berne 1996), Parr

and Yang (Parr and Yang 1989), Friesner and Berne (Stern et al. 2001), and Field

(Field 1997). The fluctuating charge model breaks the system into fragments and

for each fragment the atomic charges are obtained self-consistently by minimizing

the total energy of the system with respect to the atomic charges, subject to the

constraint that the total charge of each fragment is conserved. The resulting

charges are those for which the electronegativities (e.g., partial derivatives of the

total energy E with respect to the atomic charges) are equal. These fluctuating

charge models are also closely related to the procedure based on chemical

potential equalization, proposed by York and Yang (York 1995; York and Yang

1996) and to the Drude oscillator models implemented by Voth (Lobaugh and

Voth 1994), Chandler (Thompson et al. 1982), Cao and Berne (Cao and Berne

1992) and Sprik and Klein (Sprik and Klein 1988).

Fluctuating charge models, based on the equalization of atomic

electronegativities, have also been formulated at the semiempirical level (Borgis

and Staib 1995; Field 1997; Gao 1997). In these semiempirical methods, the self-

consistent density matrices of the constituent fragments are obtained by neglecting

the off-diagonal elements associated with atoms of different fragments.

Deficiencies in the description of the interactions between fragments have been

partially corrected with additional terms, including Lennard-Jones potentials and

electrostatic interactions based on Mulliken charges (Field 1997; Gao 1997).

Mulliken charges retain the property of relative atomic electronegativities but

provide a very poor description of the electrostatic potential around the fragment.

Therefore, some versions of these approaches have scaled the Mulliken charges

with empirical parameters obtained from comparisons of the computed and

experimental properties of the system (Gao 1997). Other semiempirical

implementations use atomic partial charges defined by the charge model (CM)

proposed by Cramer and Truhlar (Storer et al. 1995). One can see that a difference

between these empirical approaches and the MOD-QM/MM methodology is not

only the level of theory and the underlying self-consistent principle implemented

(i.e., equalization of atomic electronegativities in fluctuating charge models versus

minimization of the error in the electrostatic potential in MOD-QM/MM) but also

the fact that the MOD-QM/MM has been specifically designed to obtain ab initio

quality electrostatic potentials from first principles calculations of the atomic

6

charges (Gascon et al. 2006a; Menikarachchi and Gascon 2008; Newcomer et al.

2009). In contrast, semiempirical fluctuating charge models use atomic charges

obtained in terms of the Mulliken population analysis, or parametrized by

empirical procedures. Unfortunately, these problems of semiempirical fluctuating

charge models with regards to their limitations to model electrostatic potentials

cannot be solved by using ESP charges, as in the MOD-QM/MM method, since

the semiempirical ESP charges are generally too small in magnitude and,

therefore, would also require empirical scaling procedures (Field 1997; Gao

1997). Another important difference between MOD-QM/MM and semiempirical

fluctuating charge models is the particular treatment of the boundaries between

the QM and MM layers. MOD-QM/MM has been specifically designed to model

electrostatic potentials of extended systems (e.g., proteins) where it is necessary to

cut covalent bonds to define the boundaries between QM and MM layers.

Molecular capping fragments, in conjunction with a specific procedure for the

computation of ESP charges, makes it capable to produce ab initio quality

electrostatic potentials in extended systems.

The MOD-QM/MM method also shares common features with a variety of

other quantum chemistry methods for very large molecules, including ‘linear

scaling’ techniques where the whole system is treated at the QM level. The main

similarity is that both ‘linear scaling’ and hybrid QM/MM methods assume that

chemical phenomena are mostly local. Linear scaling-methods are subdivided into

two classes, according to how they exploit the principle of locality. One class

involves methods where the whole system is treated at once (Greengard and

Rokhlin 1987; Baldwin et al. 1993; Li et al. 1993; White and Headgordon 1994;

Strout and Scuseria 1995; Burant et al. 1996; Schwegler and Challacombe 1996;

Strain et al. 1996; Stratmann et al. 1996; White et al. 1996; Stratmann et al. 1997;

Daniels and Scuseria 1999; Babu and Gadre 2003), although at a greater

computational cost, exploiting locality to either avoid computing 2-electron

integrals between orbitals far away from each other (Strain et al. 1996), to obtain

sparse Fock matrices, circumventing the full diagonalization or to compute the

correlation energy in a less expensive way (Saebo and Pulay 1993; Ko et al.

2003), or by using localized orbitals (Subotnik and Head-Gordon 2005). ‘Divide

and conquer’ techniques belong to another class of linear scaling methods. These

approaches partition the electron density or the density matrix of a large system

7

into fragments and obtain the properties of the whole system from conventional

quantum chemical calculations of the constituent domains. Different divide and

conquer methods vary in the way the covalent bonds that join the fragments are

treated. Yang and co-workers developed the earliest divide-and-conquer method

(Yang 1991; Yang and Lee 1995; Lee et al. 1996). Merz and coworkers developed

a semiempirical version of divide-and-conquer (Dixon and Merz 1996; Dixon and

Merz 1997; Ermolaeva et al. 1999). Exner and Mezey proposed the adjustable

density-matrix assembler (ADMA) approach (Exner and Mezey 2002; Exner and

Mezey 2003; Exner and Mezey 2004), Zhang and co-workers developed the

molecular fractionation with conjugated caps (MFCC) approach (Kurisu et al.

2003; Zhang et al. 2003b; Zhang and Zhang 2003; Gao et al. 2004; Mei et al.

2005). Gadre and coworkers developed the molecular tailoring approach (MTA)

(Gadre et al. 1994; Babu and Gadre 2003; Babu et al. 2004), Kitaura and

coworkers proposed the fragment molecular-orbital sFMOd method (Kitaura et al.

1999; Kitaura et al. 2001; Fedorov and Kitaura 2004b; Fedorov and Kitaura

2004a), and Li and co-workers proposed the localized molecular orbital assembler

approach (Li et al. 2005; Li and Li 2005). These divide-and-conquer approaches

are inherently linear scaling schemes and have been demonstrated to yield quite

accurate energies and properties of entire macromolecules from calculations of the

constituent fragments. From the review of the various fragmentation methods

presented above, one can see that the MOD-QM/MM approach shares common

features with many of these earlier approaches, and is specifically designed to

provide ab initio quality electrostatic potentials and optimized structures. This is

essential for an accurate description of structural and electronic properties of

prosthetic groups embedded in macromolecules.

The MOD-QM/MM approach is not limited to any specific level of

quantum chemistry and has been applied in conjunction with Density Functional

Theory (DFT) in studies of photosystem II and other macrobiological systems

(Gascon et al. 2006a; Gascon et al. 2006b; Sproviero et al. 2006b; Sproviero et al.

2006a; Sproviero et al. 2007; Gascon et al. 2008; Menikarachchi and Gascon

2008; Sproviero et al. 2008b; Sproviero et al. 2008d; Sproviero et al. 2008a;

Sproviero et al. 2008c; Sproviero et al. 2008e). The main advantage of DFT is its

efficiency for the description of electron correlation at a computational cost

comparable to single-determinant calculations, and the efficient description of

8

metal centers and inorganic cores of metalloproteins in various possible spin-

electronic states. MOD-QM/MM calculations have modeled charge redistribution,

polarization effects and changes in the ligation scheme of metal centers induced

by oxidation state transitions, bond breaking and forming processes associated

with chemical reactivity and proton transfer. In addition, the MOD-QM/MM

method has provided a reliable description of electronic and magnetic properties

of prosthetic groups, embedded in complex molecular environments, and has

allowed for the application of post-QM/MM optimization methods based on

simulations of X-ray absorption spectra and direct comparisons with high-

resolution spectroscopic data (e.g. FT-IR, NMR, EXAFS, and EPR) (Sproviero et

al. 2007; Sproviero et al. 2008b; Sproviero et al. 2008d; Sproviero et al. 2008a;

Sproviero et al. 2008c; Sproviero et al. 2008e).

The application studies reviewed in this paper show that the MOD-

QM/MM methodology is well suited to study equilibrium and transition states of

biological systems beyond the limitations of model structures obtained by X-ray

diffraction or NMR spectroscopy, including reaction intermediates generated by

water exchange, proton transfer and the rearrangement of ligands during the

catalytic cycle of water oxidation at the oxygen-evolving complex (OEC) of

photosystem II (PSII) (Sproviero et al. 2006b; Sproviero et al. 2007; Sproviero et

al. 2008b; Sproviero et al. 2008d). The paper is organized as follows. First, we

review the QM/MM methodology, including a discussion of its specific

implementation to studies of high-valent metal complexes in metalloproteins.

Then, we review the MOD-QM/MM method and a post-QM/MM refinement

scheme based on simulations of X-ray absorption spectra and direct comparisons

with experimental measurements. The Applications Section reviews benchmark

studies illustrating the capabilities of the MOD-QM/MM methodology and the

post-QM/MM analysis of model structures through direct comparisons with NMR

and EXAFS spectroscopic data.

9

Methods

QM/MM Methodology

The MOD-QM/MM methodology (Gascon et al. 2006a; Menikarachchi and

Gascon 2008; Sproviero et al. 2008e; Newcomer et al. 2009) has been applied in

conjunction with the two-layer ONIOM Electronic-Embedding (EE) link-

hydrogen atom method (Dapprich et al. 1999), as implemented in Gaussian 03

(Frisch et al. 2004). The ONIOM approach divides the system into two partitions

(Fig. 1), including a reduced system X (the QM layer) and the rest of the system

(region Y). The total energy E is obtained from three independent calculations:

E = EMM,X+Y + EQM,X - EMM,X (1)

where EMM,X+Y is the energy of the complete system computed at the molecular

mechanics level of theory, while EQM,X and EMM,X correspond to the energy of the

reduced-system X computed at the QM and MM levels of theory, respectively.

Electrostatic interactions between regions X and Y are included in the calculation

of both EQM,X and EMM,X at the quantum mechanical and molecular mechanics

levels, respectively. Thus, the electrostatic interactions computed at the MM level

in EMM,X and EMM,full cancel and the resulting DFT QM/MM evaluation of the

total energy involves a quantum mechanical description of the polarization of the

reduced system, due to the electrostatic influence of the surrounding protein

environment.

MOD-QM/MM Method

Fig. 2 illustrates the MOD-QM/MM computational protocol where the system is

partitioned into n molecular domains by using a simple space-domain

decomposition scheme (Gascon et al. 2006a; Menikarachchi and Gascon 2008;

Newcomer et al. 2009). ElectroStatic-Potential (ESP) atomic-charges of the

constituent molecular domains are sequentially computed by using QM/MM

hybrid methods (e.g., the two-layer ONIOM-EE scheme) considering the updated

distribution of atomic charges in the molecular domains previously modeled as

QM layers in the cycle. The whole cycle of QM/MM calculations is iterated

several times until obtaining a self-consistent point-charge model of the

electrostatic potential that explicitly includes mutual polarization interactions

10

between the constituent molecular domains. The underlying computational effort

usually scales linearly with the size of the system since the iterative scheme

typically converges within a few iteration-cycles (i.e., 4 or 5 cycles). The method,

thus, bypasses the enormous demands of memory and computational resources

that would be required by a ‘brute-force’ quantum chemistry calculation of the

complete system.

The MOD-QM/MM method typically corrects the atomic charges

prescribed by popular molecular mechanics force fields (Dykstra 1993; Cornell et

al. 1995) by only 10–20%. However, summing the corrections from multiple

domains over the entire QM/MM interface often introduces significant corrections

(10-15 kcal/mol) to the total interaction between the QM and MM layers. These

energy corrections are often critical since they are comparable to energy

differences associated with different protonation states, hydrogen-bonding

structures, or the energy splitting between high-spin and low-spin states of the

metal cluster.

The accuracy of the resulting molecular electrostatic potential comes at the

expense of transferability since the atomic charges obtained for one configuration

are, in principle, nontransferable to other configurations of the system. Therefore,

while accurate, the computed electrostatic potential is most useful for applications

where the system does not involve large conformational changes, such as in

studies of quite rigid DNA quadruplexes, or intermediate structures of the OEC of

PSII where there are no significant temperature-dependent variations in structure,

protonation state, or charge localization, as recently reported by studies of X-ray

absorption at 20 K and room-temperature (Haumann et al. 2005). Other

applications that should benefit from the MOD-QM/MM method, so long as the

system does not undergo significant structural changes, include studies of enzyme

catalysis with electron transfer and proton transport (Warshel 1981; Warshel and

Aqvist 1991; Okamura and Feher 1992; Lobaugh and Voth 1996; Sham et al.

1999a; Sham et al. 1999b; Ilan et al. 2004; Popovic and Stuchebrukhov 2004), ion

channels (Aqvist and Luzhkov 2000; Luzhkov and Aqvist 2000; Bliznyuk et al.

2001; Ilan et al. 2004), docking and ligand binding (Simonson et al. 1997;

Simonson et al. 1999; Norel et al. 2001; Wang et al. 2001; Sheinerman and Honig

2002; Lappi et al. 2004; Vasilyev and Bliznyuk 2004; Cho et al. 2005; Grater et

al. 2005), macromolecular assembly (Sheinerman et al. 2000; Norel et al. 2001;

11

Wang et al. 2001), and signal transduction (Green et al. 2006; Hurley 2006;

Mulgrew-Nesbitt et al. 2006; Gentilucci et al. 2007). The wide range of potential

applications where the energetics is dominated by electrostatic energy

contributions emphasizes the importance and urgent need to establish fast, yet

accurate, methods for computations of ab initio quality electrostatic potentials in

biomacromolecular systems (Nakano et al. 2000; Kitaura et al. 2001; Exner and

Mezey 2002; Nakano et al. 2002; Exner and Mezey 2003; Exner and Mezey 2004;

Gao et al. 2004; Exner and Mezey 2005; Li et al. 2005; Szekeres et al. 2005).

MOD-QM/MM Dangling Bonds: In order to define the boundary between the

QM and MM layers in extended systems, it is necessary to cut covalent bonds

leading to the famous problem of ‘dangling bonds’. Various QM/MM methods

have been developed to deal with the electron density along the dangling bonds

according to different approximations (Lin and Truhlar 2005). The link-atom

scheme is one of the earliest methods and involves saturating the QM valency

with extra capping atoms (Singh and Kollman 1986; Field et al. 1990; Eurenius et

al. 1996). The link atom is usually a hydrogen but another atom, or group of

atoms, can be used (Antes and Thiel 1999; Zhang et al. 2003a; Chen et al. 2004;

Gao et al. 2004). Artificial halogen-like atoms called pseudohalogens have also

been used to better mimic properties such as electron donation and withdrawal

(Zhang et al. 1999). Other methods employ frozen orbitals (Gao et al. 1998;

Philipp and Friesner 1999) (Murphy et al. 2000b; Murphy et al. 2000a),

pseudopotentials (Estrin et al. 1991; Bergner et al. 1993; Zhang et al. 1999) or

electron density derived schemes (Ferre and Angyan 2002) to avoid the need to

insert extra atoms. In particular, the delocalized Gaussian MM method (Das et al.

2002) is implemented in the CHARMM/GAMESS-UK and

CHARMM/GAMESS-US software programs (Brooks et al. 1983). Karplus and

co-workers have compared link atom schemes with frozen orbital methods and

have concluded that link-atom and frozen orbital methods can give comparable

results with proper care in the partitioning and treatment of electrostatics, and

neither is systematically better than the other (Reuter et al. 2000). The MOD-

QM/MM method is not restricted to any particular treatment of the dangling

bonds at the QM/MM boundary. However, implementations with the Gaussian

program must be based on Morokuma’s ONIOM method that uses the link-

hydrogen atom scheme (Maseras and Morokuma 1995; Humbel et al. 1996;

12

Svensson et al. 1996; Dapprich et al. 1999; Vreven and Morokuma 2000a; Vreven

et al. 2001; Vreven and Morokuma 2003).

In order to use the link-hydrogen atom approach without compromising

the accuracy of the resulting electrostatic potential, MOD-QM/MM is commonly

implemented according to a molecular capping procedure where molecular

fragments are capped with their nearest neighbor fragments and the neighbors are

terminated with link-hydrogen atoms. Similar molecular capping procedures have

been investigated by Zhang and co-workers in the molecular fractionation with

conjugated caps (MFCC) approach and excellent results have been reported for

benchmark calculations (Zhang et al. 2003a; Chen et al. 2004; Gao et al. 2004). In

addition, the MOD-QM/MM approach has been implemented with the Q-site

package (Philipp and Friesner 1999; Murphy et al. 2000b; Murphy et al. 2000a), a

QM/MM program where boundaries between QM and MM layers are treated by

using frozen orbitals, instead of link-hydrogen atoms.

MOD-QM/MM ESP Charges: Computations of ESP atomic-charges for the

individual molecular fragments defined in the MOD-QM/MM method (Gascon et

al. 2006a) involve a least-squares fit procedure similar to the Besler-Merz-

Kollman method implemented in Gaussian (Besler et al. 1990), although with

different holonomic constraints. The least-squares fit criterion involves finding the

minimum of the function:

( ) ( ) ( ),,,,,,, 211

221 ∑∑ +−=

= kNkk

m

jjjN qqqgEVqqq KK λγ (2)

where jV is the ab initio electrostatic potential at a space point j and

./1

jk

N

kkj rqE ∑

=

= Constraints are imposed via the Lagrange multipliers kλ . The

Besler-Merz-Kollman method uses a single constraint, ensuring that the sum of

the atomic charges is equal to the total charge of the

fragment: ( ) ,0,,,1

211 ∑=

=−=N

ktotkN qqqqqg K where N is the number of atoms in the

QM layer. Such a fitting procedure would not be useful in the MOD-QM/MM

approach since it would leak charge to the capping fragments and, therefore,

would not conserve the total charge of the system over multiple iteration cycles.

In order to overcome this problem, the MOD-QM/MM method includes two

constraints in the least-squares fit procedure, ensuring that the sum of the charges

13

of link-atoms is equal to zero: ( ) ,0,,,1

212 ∑−

=

==MN

kkN qqqqg K where N−M is the

number of link-atoms, and that the sum of the atomic charges in the QM layer is

equal to the total charge of the fragment:( ) .0,,,1

211 ∑=

=−=N

ktotkN qqqqqg K

The MOD-QM/MM calculation of ESP charges can be described by

considering a QM layer with N atoms, where N−M are link-atoms. The total

charge of the QM layer is ,21 QQQ += where ∑−

=

+=1

11

M

iMi qqQ is the net-charge of

the fragment and ∑−

+=

+=1

12

N

MiNi qqQ is set equal to zero. The electrostatic potential

at position ,jr due to all point charges in the QM region, is ,/1

ji

N

iij rqu ∑

=

= where

.ijji rrr −= Considering the conditions imposed for 1Q and ,2Q the electrostatic

potential becomes,

.21

1

11

1 jN

N

Mi jN

i

ji

i

jM

M

i jM

i

ji

ij r

Q

r

q

r

q

r

Q

r

q

r

qu +

−++

−= ∑∑

−

+=

−

=

(3)

Making the substitutions ( )jkjijik rrF /1/1 −= and ,/1 jkjk rK = Eq. (3) can be

rewritten as follows:

.21

1

1

1

1jNjM

N

MijiNi

M

ijiMij KQKQFqFqu +++= ∑∑

−

+=

−

=

(4)

The actual computation of ESP atomic charges involves a least-squares

minimization procedure of the error function ( ) ,1

22 ∑=

−=gN

jjj Uuχ where jU is the

QM/MM electrostatic potential at grid point j and ju is the corresponding

electrostatic potential defined by the distribution of point charges. The summation

in the 2χ error function is typically carried over the set of Ng grid points

associated with four layers of grid points at 1.4, 1.6, 1.8 and 2.0 times the van der

Waals radii around the QM region. Each layer with a density of 1 grid point Å-2.

The minimum of 2χ is obtained by imposing the condition,

( ) ,021

2

∑=

=∂∂

−−=∂∂ gN

j k

jjj

k q

uUu

q

χ (5)

14

for all kq in the set ( ).,,,,, 11121 −+− NMM qqqqq KK

Eq. (4) indicates that ,jksk

j Fq

u=

∂∂

where s corresponds toM or N

depending on whether ,Mk < or ,NkM << respectively. Therefore, Eq. (5) can

be re-written as follows:

( )[ ] .1

11

1

111 21 ∑ ∑∑ ∑∑

−

+==

−

===

−=+−N

Mijks

N

j jiNi

M

i

N

j jksjiMijks

N

j jNjMj FFqFFqFKQKQU ggg (6)

Considering the complete set of charges kq , Eq. (6) is better represented in matrix

notation as follows:

,0

0

2

1

=

F

Fqc (7)

where

( )[ ] ,1 21 jks

N

j jNjMjk FKQKQUc g∑ =

+−= ( ),,,,,, 11121 −+−= NMM qqqqqq KK

,11 ∑ =

= gN

j jksjlMlk FFF and .

12 ∑ == gN

j jksjlNlk FFF Note that vector c and matrix F

are only functions of the distances between atomic positions and the grid points

jkr , the partial charges 1Q and 2Q , and the electrostatic potential evaluated at the

grid points .,,1 gNj K= This allows one to obtain the atomic-charges simply by

inversion of Eq. (7).

Charge Transfer: One of the constraints in the MOD-QM/MM calculation of

ESP charges is that the total charge of each QM domain must be conserved

(Gascon et al. 2006a). Therefore, charge-transfer corrections are included only

within molecular domains. The definition of individual molecular domains,

however, is quite flexible and can include multiple fragments involved in charge

transfer. However, charge transfer is assumed to be local within domains. This

approximation has also been used by Berne and co-workers (Rick et al. 1994) in

their simulations of liquid water, where they argued that allowing charge transfer

freely and instantaneously between molecules at large distances is likely to be

unrealistic due to kinetic restrictions. The ‘localized charge-transfer’

approximation is also fully consistent with recent studies of polarization and

charge-transfer effects in aqueous solutions (Mo and Gao 2006) that found

polarization effects to be much more important than charge-transfer corrections.

15

MOD-QM/MM Optimization

The iterative MOD-QM/MM hybrid method was initially developed for ab initio

computations of protein electrostatic potentials (Gascon et al. 2006a) and has been

recently generalized as a practical algorithm for structural refinement of extended

systems (Newcomer et al. 2009). The resulting generalized computational

protocol requires the space-domain fragmentation of the system into partially

overlapping molecular domains and an iterative cycle of QM/MM calculations as

depicted in Fig. 2. Each step in the cycle considers a different molecular domain

as a QM layer and updates its distribution of ESP atomic charges, after relaxing

its geometry by energy minimization. The rest of the system is considered at the

MM level, with an updated distribution of atomic charges for all molecular

domains previously considered as QM layers in the iteration cycle

The geometry optimization is performed with standard methods (Versluis

and Ziegler 1988; Fan and Ziegler 1991), using the Newton-Raphson method and

updating the second derivative (Hessian matrix) with the Broyden-Fletcher-

Goldfarb-Shanno strategy (Schlegel 1987). Starting with the first molecular

domain R1, its geometry is relaxed as influenced by the surrounding environment.

The ESP charges of the QM layer in its minimum energy configuration are

computed and updated for subsequent steps in the cycle (see Fig. 2). The relaxed

configuration and ESP atomic-charges of the next domain (R2) are computed

analogously, considering the updated charges and configuration of the previously

considered molecular domain. The procedure is subsequently applied to all

fragments and the entire computational cycle is iterated several times until

reaching self-consistency in the description of the geometry and distribution of

atomic charges as influenced by mutual polarization effects. The capabilities of

the resulting protocol have been recently demonstrated at the ONIOM-EE level,

as applied to the description of benchmark models systems that allowed for direct

comparisons with full QM calculations as well as in the study of the structural

characterization of the DNA Oxytricha nova guanine quadruplex (G4) (Newcomer

et al. 2009).

ONIOM-EE Optimization: During the MOD-QM/MM optimization procedure,

the coupling between the QM and MM layers includes both bonded interactions

(e.g., bond stretching, bond bending, and internal torsions) as well non-bonding

interactions (e.g., electrostatic and Van der Waals interactions). Electrostatic

16

terms often dominate the interactions between the QM and MM layers. In the

electronic embedding (EE) scheme (Vreven and Morokuma 2000b; Vreven et al.

2003; Vreven et al. 2006), the QM computations are carried out by including

perturbation terms that describe the electrostatic interactions between the QM and

MM layers as one-electron operators of the QM Hamiltonian. The electrostatic

potential due to the MM layer is conveniently represented by atomic-centered

partial point charges in the effective QM Hamiltonian. More complicated

expansions, including distributed multipoles (Ferre and Angyan 2002) could also

be applied but so far have not been implemented in conjunction with the MOD-

QM/MM method. Bonding and non-bonding Van der Waals interactions between

fragments in the two layers are retained at the MM level.

In order to explain the underlying optimization process in MOD-QM/MM,

as implemented in conjunction with ONIOM, we consider the gradient of the

ONIOM energy

JJ ⋅∂

∂−∂

∂+⋅∂

∂=∂

∂ +

q

E

q

E

q

E

q

E MMX

MMYX

QMX

ONIOM

(8)

where J is the Jacobian that converts the coordinates of the reduced system (X)

into the coordinates of the complete system (X+Y). When the coordinates of X

(qX) are frozen, the gradient of the energy becomes a function of MMYXE + only. Thus,

it is possible to fully minimize the energy with respect to the coordinates in the

MM layer (qY) by simply using MM level calculations (i.e., microiterations). Such

an optimization process is followed by a QM energy minimization step that

updates the coordinates of the reduced system by using the forces calculated

according with Eq. (8). The process of microiterations, followed by the QM

optimization of the reduced system, is repeated until the ONIOM energy has

converged to a minimum with respect to all nuclear coordinates.

In the ONIOM electronic-embedding (EE) method, the coordinates of the

system cannot be rigorously separated into qX and qY, since 0/ ≠∂∂ YQMXE q and

0/ ≠∂∂ YMMXE q . Therefore, the microiteration scheme outlined above is no longer

valid. To address this problem, several modified schemes have been proposed

(Vreven et al. 2003). In particular, the ONIOM-EE optimization scheme

implemented in Gaussian 03 approximates the electrostatic interactions between

QM and MM layers during the microiterations as described by the distribution of

17

ESP charges computed for the initial configuration. After completing the

microiterations with essentially a frozen distribution of atomic charges, the wave

function of the QM layer is reevaluated and new ESP charges are obtained for

subsequent microiterations. The QM wave function is recomputed and the MM

geometry is reoptimized several times until reaching self-consistency. Once self-

consistency is reached, a QM energy minimization step updates the coordinates of

the QM layer and the entire process is repeated until convergence.

MOD-QM/M Model of Photosystem II: MOD-QM/MM models of the OEC of

PSII have been based on the 1S5L X-ray crystal structure of PSII from the

cyanobacterium Thermosynechococcus elongatus 1S5L (Ferreira et al. 2004) (see

Fig. 3). The models explicitly considered 1987 atoms of PSII, including the

proposed Mn3CaO4Mn unit and all amino acid residues with α -carbons within 15

Å from any atom in the OEC metal ion cluster. The coordination spheres of the

Mn centers were completed by hydration, assuming a minimum displacement of

the proteinaceous ligands from their crystallographic positions. The usual

coordination of 5 or 6 ligands, for high-valent Mn ions with oxidation states III

and IV, was also satisfied. In addition, the coordination of calcium (typically with

6–8 ligands) was satisfied by coordination of water molecules and negative

counterions.

The MOD-QM/MM molecular structures were hydrated by including water

molecules that could bind to polar amino acid residues, or existing water molecules

in the model. The hydration process involved inserting the model system in a large

box containing a thermal distribution of water molecules. After adding to the model

the water molecules that did not interfer sterically with the protein residues, the

structure of hydrogen bonds was optimized by energy minimization. The hydration

and relaxation processes were repeated several times until the number of water

molecules in the system converged, typically with the addition of 85–100 water

molecules. A few of the additional water molecules (up to six molecules) attached to

metal centers in the cuboidal Mn3CaO4Mn cluster, including 2 waters molecules that

were considered as potential substrate water molecules responsible for O-O bond

formation in the S4 to S0 transition. One of the water molecules is bound to Ca2+ as

suggested by 18O isotope exchange measurements (Hillier and Wydrzynski 2004;

Hillier and Wydrzynski 2008). The resulting hydrated structure of the

oxomanganese cluster is roughly consistent with pulsed EPR experiments revealing

18

the presence of several exchangeable deuterons near the Mn cluster in the S0, S1, and

S2 states (Britt et al. 2004).

MOD-QM/MM computations of the OEC of PSII could only be efficiently

performed, according to the ONIOM-EE level of theory, after obtaining high-

quality initial-guess states of the QM layer. These correspond to spin-electronic

states of the cluster of Mn ions based on ligand field theory as implemented in

Jaguar 5.5 (Jaguar 5.5; Schroedinger). Once the initial state was properly defined,

the combined approach exploited important capabilities of ONIOM as

implemented in Gaussian 03 (Frisch et al. 2004), including both the link-

hydrogen atom scheme for efficient and flexible definitions of QM layers and

the possibility of modeling open-shell systems by performing unrestricted DFT

(e.g. UB3LYP) calculations in a variety of spin-electronic states.

The QM layer of MOD-QM/MM calculations (region X) included the

Mn3CaO4-Mn complex and the directly ligated proteinaceous carboxylate groups

of D1-D189, CP43-E354, D1-A344, D1- E333, D1-D170, D1-D342. In addition,

the imidazole ring of D1- H332 as well as bound water molecules, hydroxide and

chloride ions were included in the QM layer. The rest of the system defined the

MM layer (region Y). The boundary between the QM and MM layers was defined

by cutting covalent bonds in the corresponding amino acid residues (i.e., D1-

D189, CP43-E354, D1-A344, D1- E333, D1-D170, D1-D342, and D1-H332) and

completing the covalency of frontier atoms according to the standard link-

hydrogen atom scheme (see Fig. 1).

The efficiency of MOD-QM/MM calculations has been optimized by

using a combination of basis sets for the QM layer, including the lacvp basis set

for Mn ions in order to consider electron core potentials, the 6-31G(2df) basis set

for bridging O2- ions to include polarization functions on µ-oxo bridging oxides,

and the 6-31G basis set for the rest of the atoms in the QM layer. Such a

combination of basis sets has been validated through extensive benchmark

calculations on high-valent manganese complexes (Sproviero et al. 2006a). The

molecular structure beyond the QM layer is described by the Amber MM force

field (Cornell et al. 1995).

Fully relaxed MOD-QM/MM model structures were obtained by energy

minimization of the QM layer, surrounded by the MM layer of amino acid

residues with α -carbons within 15 Å from any atom in the OEC metal cluster, in

19

the presence of a buffer shell of amino acid residues with α -carbons within 15-20

Å from any atom in the OEC ion cluster. The buffer shell is subject to harmonic

constraints in order to preserve the natural shape of the system even in the absence

of the membrane.

The electronic states of the Mn cluster involve antiferromagnetic couplings

between manganese centers and are defined by broken-symmetry (BS) states

(Noodleman 1981; Noodleman and Davidson 1986; Noodleman and Case 1992;

Noodleman et al. 1995). The BS approach, in conjunction with DFT, is the most

accurate approach available for these types of high-valent metal clusters. BS-DFT

also facilitates the ligand field analysis, including metal d d, charge transfer

(ligand metal, metal ligand), and intervalence charge transfer (metalmetal or

ligand ligand) transitions. The QM layer is typically prepared in various

possible spin states as stabilized by specific arrangements of ligands. The purity

of such a state is preserved throughout the geometry optimization process so long

as the electronic state is compatible with the geometry of the Mn3CaO4Mn cluster

and the specific arrangement of ligands. Otherwise, states of different symmetry

become more stable and the nature of the electronic state changes during the

optimization process. Upon convergence, the resulting optimized structures are

analyzed and assessed not only in terms of the total energy of the system but also

in terms of the comparison of structural, electronic and mechanistic features that

should be consistent with experimental data. Such a comprehensive analysis is

crucial for these complex biological systems often with many possible

configurations with similar energies (e.g., within the ~5 kcal/mol range of error of

DFT QM/MM calculations).

Post-QM/MM Structural Refinement Based on Spectrosc opic Data

The typical energy landscape of macrobiological systems is very rugged, with a

manifold of configurations of comparable energy (e.g., within the ~5 kcal/mol

energy range of error of DFT calculations). Therefore, it is rather difficult to

determine the most likely configuration of the system solely from the energetic

analysis of possible configurations. In order to address these limitations, a post-

QM/MM structural refinement scheme has been introduced to assess the

likelihood of structures of similar energies in terms of simulations of high-

20

resolution spectroscopy (e.g., EXAFS spectra) and direct comparisons with

experimental data (Sproviero et al. 2008c). The method is particularly valuable to

refine the configuration of prosthetic groups with multiple high-valent metal

centers beyond the current limitations of DFT-QM/MM methods and has been

successfully applied to the development of structural models of the OEC of PSII.

The refinement algorithm starts with a reference configuration of the

system (e.g., the MOD-QM/MM minimum energy structure) and iteratively

adjusts its configuration to minimize the mean square deviation between the

simulated and experimental EXAFS spectra, subject to the constraint of minimum

displacements relative to the reference configuration (see Fig. 4). Upon

convergence, the resulting structures are often consistent with the reference MOD-

QM/MM minimum energy structures (with energies and root-mean-square

deviations within the range of error of DFT QM/MM minimization methods) and

predict high-resolution spectroscopic data in quantitative agreement with

experiments. The refinement scheme allows one to go beyond the limitations of

the energetic analysis in order to explore a manifold of structures with energies

close to the energy of the MOD-QM/MM minimum energy configuration (e.g.,

within the ~5 kcal/mol DFT error) and narrow the range of possible structures

through direct comparisons with high-resolution spectroscopic data. As mentioned

earlier, this method is particularly valuable for metalloproteins for which the most

accurate and reliable experimental information on the structure of the prosthetic

group metal clusters comes from high-resolution spectroscopy (e.g., EXAFS).

The MOD-QM/MM refinement protocol has been applied in conjunction

with simulations of EXAFS spectra for the development of models of the OEC of

PSII (Sproviero et al. 2008c). The EXAFS spectra were computed as a function of

the nuclear coordinates and compared to the experimental spectra. The nuclear

coordinates of the Mn cluster and coordinated ligands were adjusted to minimize

the mean-square deviation ( )∑ −=N

ii

calci AA

N

2exp2 1χ , where calciA and exp

iA

correspond to the calculated and experimental EXAFS amplitudes for a grid of

electron kinetic energies iK with Ni −= 1 . The 2χ minimization is performed

subject to the constraints of minimum displacements of nuclear positions relative

to their coordinates in the reference MOD-QM/MM configuration. The

minimization algorithm implements a multidimensional variable metric method

21

through the Broyden-Fletcher-Goldfarb-Shanno (BFGS) approach (Broyden 1970;

Fletcher 1970; Goldfarb 1970; Shanno 1970), using the Cartesian coordinates xi of

the nuclei as input variables and an external simulation procedure that returns the

corresponding spectrum as computed by the program FEFF8 (version 8.2)

(Ankudinov et al. 1998; Bouldin et al. 2001; Ankudinov et al. 2002).

It is important to note that while the post-QM/MM structural refinement

algorithm depicted in Fig. 4 has been implemented in terms of simulations of

EXAFS spectra, the2χ minimization procedure could be based on other

spectroscopic properties of the system if experimental data are available and the

properties can be readily computed. Examples include structural refinement

schemes based on the optimum description of magnetic properties of specific

centers as directly compared to experimental EPR data, or NMR chemical shifts.

Refinement schemes for ligand binding configurations could be based on direct

comparisons of calculated binding enthalpies and microcalorimetry data. In

addition, refinement schemes could optimize a set of generalized coordinates (as

in rigid docking techniques) rather that a set of nuclear Cartesian coordinates.

These generalized coordinates can be selected to address difficult problems such

as the compensation for systematic errors (both in the computed and measured

properties), or global reference values such as the Fermi energy in EXAFS

calculations, or the value of the (external or internal) reference in calculations

NMR chemical shifts.

EXAFS Simulations

Ab initio simulations of EXAFS spectra involve solving the multiscattering

problem associated with the photodetached electrons emitted upon X-ray

absorption by the Mn centers. The quantum mechanical interference of outgoing

photoelectrons with the backscattering waves from atoms surrounding the Mn

ions, gives rise to oscillations of EXAFS intensities as a function of the kinetic

energy (or momentum) of the photoelectron. The Fourier transform of these

oscillations provide information on the Mn-Mn distances, the coordination of Mn

ions, and changes in the Mn coordination sphere as determined by changes in the

oxidation state of the metal centers. Calculations are typically carried out

according to the Real Space Green function approach as implemented in the

22

program FEFF8 (version 8.2) (Ankudinov et al. 1998; Bouldin et al. 2001;

Ankudinov et al. 2002), using the theory of the oscillatory structure due to

multiple-scattering originally proposed by Kronig (Kronig 1931; Kronig and

Penney 1931) and worked out in detail by Sayers (Sayers et al. 1971), Stern

(Stern 1974) Lee and Pendry (Lee and Pendry 1975), and by Ashley and Doniach

(Ashley and Doniach 1975).

One of the advantages of the structural refinement scheme based on

EXAFS simulations and comparisons with experimental data is that EXAFS

experiments require doses of X-ray radiation that are substantially lower than

those required by X-ray diffraction (XRD) measurements (Yano et al. 2005). The

onset of radiation damage can be precisely determined and controlled by

monitoring the absorber metal K-edge position, thus allowing comparisons with

spectroscopic data from an (almost) intact metal cluster within the protein. In

addition, EXAFS provides metal-to-metal and metal-to-ligand distances with high

accuracy (~0.02 Å) and a resolution of ~0.1 Å. For polarized EXAFS experiments

on one-dimensionally oriented membranes or three-dimensionally oriented single

crystals, the EXAFS amplitude is orientation dependent and proportional to

)(cos2 θ , where θ is the angle between the electric field vector of the polarized

X-ray beam and the absorber back-scatterer vector (see Fig. 5). Therefore, the

structural refinement can provide additional geometric information about the

metal site in metalloprotein crystals (Scott et al. 1982; Flank et al. 1986; Yano et

al. 2005).

Post-QM/MM Refinement Model of the OEC of PSII: In recent studies, a

structural model of the OEC of PSII has been developed (see Fig. 3) by using the

post-QM/MM structural refinement scheme in terms of simulations of isotropic

and polarized EXAFS spectra and direct comparisons with experimental data

(Sproviero et al. 2008c). The MOD-QM/MM minimum energy structure of the

OEC of PSII in the S1 state was used as the initial reference configuration. Upon

convergence of the structural refinement procedure, the isotropic and polarized

EXAFS spectra of the resulting refined (R)-QM/MM model were found to be in

quantitative agreement with experimental data (see Figs. 6 and 7).

Figs. 6 and 7 show that the R-QM/MM structure (see Fig. 8) is consistent

with high-resolution EXAFS measurements (Haumann et al. 2005; Yano et al.

2005) as well as with the description of the protein environment suggested by

23

XRD models (Ferreira et al. 2004; Loll et al. 2005). The predicted metal-metal

and metal-ligand distances are consistent with EXAFS studies of frozen solutions

of PSII that have provided accurate distances for Mn-Mn, Mn-Ca, and Mn/Ca-

ligand vectors in the Mn4Ca cluster of PSII (Dau et al. 2001; Robblee et al. 2001;

Haumann et al. 2005; Yano et al. 2005). In addition, the R-QM/MM model

provides insights on structural details currently unresolved by XRD spectroscopy,

including the spatial orientation of the cluster as well as the approximate

placement of the models within the protein environment. However, it is important

to emphasize that the R-QM/MM model does not rule out other possible structures

since it is not the result of an exhaustive refinement of all possible structures of

the system but rather a local minimum solution relative to the reference MOD-

QM/MM structure.

The oxomanganese inorganic core of the R-QM/MM model involves a

cuboidal core Mn3CaO4 with a dangler manganese ligated to a corner µ4-oxide

ion, partially consistent with previous proposals (Britt et al. 2000; Dau et al. 2001;

Robblee et al. 2001; Dau et al. 2004; Ferreira et al. 2004; Sproviero et al. 2006b;

Sproviero et al. 2006a; Sproviero et al. 2007; Sproviero et al. 2008b; Sproviero et

al. 2008d; Sproviero et al. 2008a; Sproviero et al. 2008e). In particular, the

cuboidal core is similar to the core of the XRD model of PSII from the

cyanobacterium Thermosynechococcus elongatus 1S5L (Ferreira et al. 2004) that

tentatively proposed positions for the Mn ions and oxo bridges. The model is

partially consistent with the overall electronic density maps, with the metal-metal

distances determined by isotropic EXAFS studies (Yachandra et al. 1996), and

with the structure of µ-oxo bridges linking the metal centers as suggested by

earlier proposals (Britt et al. 2000).

Dangler Mn models were originally proposed (Britt et al. 2000; Britt et al.

2004) within a set of 11 possible structural motifs, suggested by solution EXAFS

studies, from which the dimer-of-dimers model was highly favored (Britt et al.

2000; Yachandra 2002; Britt et al. 2004). Similar dangler Mn models were also

preferred by 55Mn electronuclear double resonance (ENDOR) studies that strongly

disfavored the dimer-of-dimers motif over models with a trinuclear Mn core and a

fourth Mn set off from the core by a longer Mn-Mn internuclear distance (Britt et

al. 2000). However, recent polarized-EXAFS studies of single crystals have

disfavored dangler models (Yano et al. 2006) and have suggested instead four

24

alternative models (models I, II, IIa and III (Yano et al. 2006)) for which the

simulated EXAFS spectra (in the absence of ligands) match the experimental data.

These empirical models, however, remain uncertain since structural differences

between some of them (e.g., models I and II) are large. Furthermore, placing any

of the four models into the XRD structures of PSII results in unsatisfactory metal-

ligand distances, coordination numbers and geometries. The underlying

inconsistencies might be due to the coordinate error in the X-ray crystal models,

or due to the intrinsic limitations of polarized-EXAFS models built by neglecting

the contributions of electron scattering paths from the ligands (i.e., assuming that

the EXAFS amplitudes result solely from scattering paths associated with metal

centers and oxo-bridges in the inorganic core). These issues should be addressed

in the development of empirical models since calculations of polarized-EXAFS

spectra suggest that the contributions of backscattering from the ligands could

introduce amplitude corrections of the order of 10–15 % (Sproviero et al. 2008c).

Exchange coupling constants

The post-QM/MM analysis of MOD-QM/MM structural models can be based on

calculations of magnetic properties and direct comparisons with experimental

data. Biomimetic oxomanganese complexes (Cooper and Calvin 1977; Cooper et

al. 1978; Stebler et al. 1986; Sarneski et al. 1990; Thorp and Brudvig 1991;

Manchanda et al. 1994; Limburg et al. 1999; Ruettinger et al. 1999) and models of

the Mn4Ca complex of PSII extracted from complete MOD-QM/MM models

(Sproviero et al. 2006b) have been recently analyzed (Sproviero et al. 2006a). The

exchange coupling constants Jij between centers i, j were computed assuming that

the centers interact according to the Heisenberg Hamiltonian,

∑−=N

jijiij SSJH

,

ˆˆ , (9)

where Si is the spin corresponding to metal center i. In the weak-coupling limit,

the spin states are essentially localized on individual metal centers and the

exchange coupling constants are determined by splittings between energy levels

as follows,

( )jiji

ijHS

ijBS

ij SSSS

EEJ

,min2

)()(

+−= , (10)

25

where )(ijBSE and )(ij

HSE are the magnetic energy contributions from the pair of metal

centers i and j (with Si < Sj) in the broken-symmetry (BS) and high-spin (HS)

configurations, respectively. The HS configuration corresponds to equally

oriented spins while in the BS state the α and β densities are allowed to localize

on different atomic centers, providing a multiconfigurational character to the spin

state (Noodleman 1981; Noodleman and Davidson 1986; Noodleman and Case

1992; Noodleman et al. 1995).

The physical picture supporting this calculation is that in the weak-

coupling limit, atomic d-orbitals remain singly occupied and the spin states are

essentially localized on individual metal centers. Under those conditions, the

ground state of a multinuclear complex involves anti-parallel coupling between

spins localized on different centers. However, metal-metal interactions induce

electronic delocalization over multiple centers, forming bonding and antibonding

orbitals by either direct overlap of atomic orbitals or superexchange. In the BS

state, α and β electronic densities can be localized on different centers. These

relaxed symmetry requirements allow for substantial interactions mediated by d-

orbitals of the metal centers and the p-orbitals of the ligands.

To perform ab initio calculations of J, it is convenient to define spin

configurations that differ only in the orientation of the spin of one center with

respect to the HS configuration. The energy difference associated with flipping a

spin of center k with respect to the HS configuration is,

( )( )∑

∑

≠

≠

+−=

−=−

N

kjkjkjjk

N

kj

jkBS

jkHSSCkHS

SSSSJ

EEEE

,min2

)()(

(11)

where k = 1, ..., N. Considering that there are N + 1 relevant spin configurations,

including one with the highest spin and N with a spin flipped in only one center,

the coupling constants can be readily obtained since the number of equations is

equal or greater than the number of unknown Js.

Benchmark Calculations: Benchmark calculations of exchange coupling

constants involved the analysis of of di-, tri- and tetra-nuclear Mn complexes

(Sproviero et al. 2006a), previously characterized both chemically and

spectroscopically (Cooper and Calvin 1977; Cooper et al. 1978) (see Table 1).

These include the di-µ-oxo bridged dimers 1: [MnIVMnIII (µ-O)2-(H2O)2(terpy)2]3+

26

(terpy = 2,2’:6,2’’-terpyridine) and 2: [MnIIIMnIV(µ-O)2(phen)2]3+ (phen = 1,10-

phenanthroline), and the trimer 3: [Mn3O4(bpy)4(OH2)2]4+ (bpy = 2,2’-bipyridine),

shown in Fig. 9. These synthetic complexes have been previously characterized

by electrochemical and high-resolution EPR and X-ray diffraction methods

(Cooper and Calvin 1977; Cooper et al. 1978).

Table 1 compares exchange-coupling constants for synthetic

oxomanganese complexes 1-3 to readily available experimental data (Cooper and

Calvin 1977; Cooper et al. 1978; Stebler et al. 1986; Sarneski et al. 1990).

Exchange coupling constants were obtained at the DFT/B3LYP level and show

very good agreement with experimental data. These results serve to validate the

analyzed model structure and its electronic description, as applied to the

description of magnetic properties of oxomanganese complexes with pre-defined

spin-electronic states. The reported agreement between calculated and

experimental exchange coupling constants was consistent with anti-ferromagnetic

couplings of high-valent metal centers in high-spin states. The analysis of the

electronic structures show that the most important exchange interactions between

Mn centers involved symmetric super-exchange pathways xzxzJ / and yzyzJ /

mediated by the overlap of orbitals xzd and yzd of Mn with out-of-the-plane πp

orbitals of the µ-oxo bridges (Sproviero et al. 2006a). These results are

particularly relevant to the description of active sites of metalloproteins with

common prosthetic groups and provide insight on the origin of electronic

delocalization over multiple centers.

Vibrational Frequencies

The assessment of MOD-QM/MM structural models has also been based on

calculations of the vibrational frequencies of the metal ligands and direct

comparisons to high-resolution FTIR measurements (Debus et al. 2005; Strickler

et al. 2005; Strickler et al. 2006). One of the questions addressed by these

comparative studies has been the ligation of the C-terminus of the D1-polypeptide

to the metal center of the OEC of PSII (Gascon et al. 2008). The MOD-QM/MM

models position the -COO− group of D1-Ala344 near calcium consistently with

XRD models (Ferreira et al. 2004). However, the S2-minus-S1 FTIR difference

spectra of PSII core complexes have been interpreted to rule out calcium ligation

27

since they show a red-shift in the symmetric stretch of the -COO− of D1-Ala344

but no change when Sr2+ is substituted for Ca2+ (Strickler et al. 2005). This

apparent contradiction between FTIR and QM/MM models has been addressed in

terms of the vibrational analysis of the α-COO− group of D1-Ala344 in MOD-

QM/MM models of the OEC of PSII in the S1 and S2 states as well as in models

where Sr2+ is substituted for Ca2+.

The MOD-QM/MM models of PSII suggest that the symmetric stretch

vibrational frequency of the α-COO- group of D1-Ala344 should changes from

1381 cm-1 in the S1 state to 1369 cm-1 in the S2 state (∆νsym(COO-) = -12 cm-1),

upon S1 to S2 oxidation of the OEC of PSII. This vibrational frequency-shift is

comparable to FTIR data although the C-terminus is coordinated to calcium, as

suggested by X-ray diffraction models. These results suggest that ligation of the

C-terminal carboxylate to calcium is consistent with both FTIR and X-ray

diffraction experiments. The observed frequency-shift is most likely caused by

oxidation of the OEC, even when the carboxylate ligand is not directly

coordinated to the oxidized Mn center. The molecular origin of the frequency shift

may be attributed to the underlying redistribution of charge induced by the S1–S2

transition. This oxidation changes the overall charge of the OEC from neutral in

the S1 state to +1 a.u. in the S2 state and the accumulation of charge reinforces

charge transfer interactions between the COO- group of Ala344 and the positively

charged metal complex. In addition, these interactions induce delocalization of

negative charge from the Ala344 carboxylate into the metal cluster, partially

neutralizing the increase of positive charge due to oxidation of the Mn complex

(Fig. 10) (Gascon et al. 2008). As a result, there is a net decrease in the bond

orders of the α-COO- bonds in the D1-Ala344 that exhibit the largest

displacement along the symmetric stretch mode.

Consistent results were obtained when analyzing reduced models of

hydrated high-valent manganese ions where an alanine amino acid is coordinated

to Ca2+ in close contact with a hydrated manganese ion undergoing Mn3+ to Mn4+

oxidation. In addition, the analysis of these models where alanine is coordinated

to Ca2+ (or Sr2+) has shown that the substitution of Ca2+ by Sr2+ produces only

minor changes in the vibrational frequency of the symmetric stretch mode of Ala

(e.g., shifting the symmetric stretch frequency from 1390 to 1391 cm-1 in the Mn3+

state and from 1350 to 1351 cm-1 in the Mn4+ state).

28

This vibrational analysis of MOD-QM/MM models in conjunction with

studies of hydrated complexes has provided valuable insights on the interpretation

of high-resolution FTIR measurements. The analysis is consistent with the

observation that the substitution of Ca2+ by Sr2+ in PSII produces negligible

changes in the vibrational frequencies of D1-Ala344 while oxidation state

transitions can induced vibrational frequency shifts, due to the redistribution of

charge in the complex, even on amino acid residues that are not directly

coordinated to the oxidized metal center.

Water Exchange Rates

The MOD-QM/MM methodology has been recently applied to develop structural

models of transition-state structures the OEC of PSII generated by water

exchange, in effort to provide insights on the nature of water exchange rates, and

rigorous validations of the structural models by direct comparisons of water-

exchange rates with time-resolved mass spectrometry (MS) measurements

(Sproviero et al. 2008e).

Time-resolved MS studies have determined different water exchange rates

(kex) of two substrate water molecules of the OEC with bulk 18O-labelled water.

The more slowly exchanging water (Wslow) was found to be associated with Ca2+,

implying that the fast-exchanging water (Wfast) would be bound to a manganese

ion (Messinger et al. 1995; Hillier and Wydrzynski 2000; Hillier and Wydrzynski

2001; Hendry and Wydrzynski 2002; Hendry and Wydrzynski 2003; Hillier and

Wydrzynski 2004). These findings were rather surprising since high-valent

manganese ions (e.g. Mn3+ or Mn4+) were expected to bind water more strongly

than Ca2+. In addition, it was observed that the exchange rate of Wslow (kex slow)

increased by two orders of magnitude upon S1 to S2 oxidation, with kex(S1) =

0.02 s-1 and kex(S2) = 2.0 s-1 (Hendry and Wydrzynski 2003; Hillier and

Wydrzynski 2004). These rates correspond approximately to activation barriers of

20 and 17 kcal/mol, respectively. This faster exchange of Wslow in the S2 state was

also intriguing since it implied that the oxidation of a manganese center should

indirectly affect the exchange rate of a calcium-bound water molecule.

The QM/MM structural models allowed for a quantitative analysis of

specific metal-water interactions and potential-energy profiles associated along

29

minimum energy paths (MEPs) for water detachment and exchange (Sproviero et

al. 2008e). The MEPs were found by energy minimization with respect to nuclear

and electronic coordinates, while progressively detaching substrate water

molecules from their corresponding coordination metal centers. Since elongation

of the metal-oxygen bond is the primary step in water exchange, and presumably

rate determining in this case, the resulting structural rearrangements provided

insights on the detailed mechanisms of water exchange.

The MOD-QM/MM models are consistent with coordination of substrate

water molecules as terminal ligands, in agreement with earlier proposals

(Hoganson and Babcock 1997; Haumann and Junge 1999; Schlodder and Witt

1999; McEvoy and Brudvig 2004; Messinger 2004; Sproviero et al. 2006b;

Sproviero et al. 2007), although in contrast to other suggested models that involve

coordination as oxo-bridges between Mn ions (Brudvig and Crabtree 1986;

Pecoraro et al. 1994; Yachandra et al. 1996; Nugent et al. 2001; Robblee et al.

2001; Messinger 2004). The energy barriers for water exchange in the MOD-

QM/MM models were estimated to be 19.3, 16.6, 8.4 and 7.9 kcal/mol for

exchange of water ligands from Ca2+(S1), Ca2+(S2), Mn(4)3+(S2) and Mn(4)3+(S1),

respectively. Figs. 11-13 show the initial and TS configurations along the MEPs

for water exchange from Ca2+ and Mn(4), as described by the DFT-QM/MM

structural models of the OEC, including a detailed description of coordination

bond lengths. In the initial states (Fig. 11), substrate water molecules are shown

attached to the metal centers forming hydrogen bonds with the exchanging water

molecules. In the TS configurations (Figs. 12 and 13), the coordination bonds of

the substrate water molecules are stretched and the substituting water molecules

are displaced relative to their original configurations. Further displacement

beyond the transition state leads to coordination of the exchanging water molecule

to the metal center. The underlying process involves converted exchange with

dissociative character (i.e. without formation of any intermediate state of lower or

higher coordination number).

The comparison of the coordination bond lengths of water ligands in TS

configurations indicates that the coordination bonds are more stretched when

water is exchange from Ca2+ in the S2 state, suggesting that the exchange

mechanism gains dissociative character upon S1/S2 oxidation of the OEC. The

electronic origin of these differences has been traced to the redistribution of

30

charge induced by charge-transfer interactions with D1-A344 leading to a net

reduction of the atomic charge of Ca2+. The electrostatic interactions with

negatively charged ligands are strenghen upon S1 to S2 oxidation since the metal

complex becomes positively charged. Most of the electronic density acquired by

Ca2+ upon S1/S2 oxidation (∆q = −0.21) is transferred from the ligated carboxylate

group of D1-A344. These results suggest that oxidation state transitions of the

oxomanganese complex can indirectly regulate substrate water binding to the

metal cluster by modulating charge-transfer interactions.

Water exchange in the coordination sphere of the dangling manganese is

also predicted to have dissociative character in both the S1 and the S2 states,

probably due to the reduced charge of the metal center in the presence of charge

transfer interactions between Mn and oxo-bridges. The calculated energy barriers

for water exchange from Mn(4) (7.9 and 8.4 kcal/mol in the S1 and S2 states,

respectively) are comparable to the barriers in hydrated Mn complexes, such as

[(H2O)2(OH)2 MnIV(µ-O)2MnIV(H2O)2(OH)2] and [MnIV(H2O)2(OH)4], where the

exchange of terminal water ligands requires 8.6-9.6 kcal/mol (Tsutsui et al. 1999;

Lundberg et al. 2003). In contrast, the DFT-QM/MM energy barriers for water

exchange from Ca2+ are much higher (19.3 and 16.6 kcal/mol for the OEC in the

S1 and S2 states, respectively). These larger barriers stabilizing substrate water

molecules bound to the catalytic metal center for much longer times (ms) than

typically bound to similar complexes in solutions.

The analysis of MOD-QM/MM models suggests that the higher energy

barriers are mainly determined by the nature of the interactions in the OEC-

binding pocket and the incomplete solvation of dissociated water ligands. These

calculations, thus, provided theoretical support for the rather slow exchange rates

and for the surprising experimental finding that the slow-exchanging substrate

water is associated with calcium, while the fast-exchanging substrate water is

coordinated with manganese. Furthermore, the QM/MM models provided a

rationale for the opposite effect of the S1/S2 transition on the two water-exchange

rate constants. It was concluded that the mechanism and energetics of water

binding to metal centers in the OEC is not simply correlated to the formal

oxidation states of the ions but rather to their corresponding partial charges, as

modulated by charge-transfer interactions with coordinated ligands.

31

MOD-QM/MM Models of other Macromolecules

The MOD-QM/MM method has been applied to a variety of systems that allowed

for direct comparisons with experimental data, including NMR, EXAFS, X-ray

diffraction and calorimetry measurements (Gascon et al. 2006a; Sproviero et al.

2006b; Sproviero et al. 2006a; Sproviero et al. 2007; Gascon et al. 2008;

Menikarachchi and Gascon 2008; Sproviero et al. 2008b; Sproviero et al. 2008d;

Sproviero et al. 2008a; Sproviero et al. 2008c; Sproviero et al. 2008e). In addition,

benchmark studies of several small model systems allowed for validation of the

mod-QM/MM method as directly compared to ’brute-force’ ab initio calculations

(Gascon et al. 2006a).

Recent efforts have been focused on benchmark studies of the MOD-

QM/MM method as applied to well-characterized model systems, including the

DNA Oxytricha nova guanine quadruplex (Fig. 14) for which high-resolution X-

ray diffraction (Haider et al. 2002; Haider et al. 2003) and NMR stuctures (Gill et

al. 2006) are available. The study aimed to explore structural models that would

predict NMR chemical shifts in quantitative agreement with experimental data, as

modulated by electrostatic interactions due to embedded monovalent cations

(Newcomer et al. 2009). Comparisons of calculated and experimental 1H NMR

chemical shifts were focused on the imino and aromatic hydrogens, shown in Fig.

15, since the magnetic properties of these centers are highly sensitive to

hydrogen-bonding and stacking interactions determining the overall configuration

of the system. The proper description of electrostatic interactions, including

polarization effects, was crucial since the metal (M+) ions (M+=Tl+, or K+)

embedded in the quadruplex polarize the guanine moieties and modulate the

strength of hydrogen bonds and stacking interactions that affect the 1H NMR

chemical shifts. The correlation between experimental and calculated chemical

shifts, shown in Fig. 16, illustrates the significant improvement in computations of

observables, as compared to the description of electrostatic interactions provided

by a standard molecular mechanics force field (Newcomer et al. 2009).

The MOD-QM/MM model of the DNA quadruplex was prepared from the

X-ray crystal structure (PDB code 1JPQ) (Haider et al. 2002). The model was

completed with hydrogen atoms, fully hydrated, and fragmented into multiple

molecular domains by using a space-domain decomposition scheme. The

fragmentation scheme was based on partially overlapping molecular domains,

32

defined as pairs of guanine residues within quartets, to ensure that every hydrogen-

bond is considered within a QM layer during its geometry optimization (Fig. 17).

The geometry of the complete system was optimized according to the MOD-

QM/MM energy minimization cycle, shown in Fig. 2. Only nucleotides involved

in hydrogen bonding interactions were allowed to relax.

The 1H NMR NMR spectra of optimized MOD-QM/MM structures were

computed at the DFT QM/MM (BH&H/6-31G*:Amber) level by using the gauge

independent atomic orbital (GIAO) method for the ab initio self consistent-field

(SCF) calculation of NMR chemical shifts implemented in Gaussian 03. Extended

QM layers, including guanine moieties in the planes above and below the

nucleotide whose chemical shifts were computed were necessary to account for

the magnetic flux associated with stacking interactions.

Conclusions

A wide range of computational methods has been developed during the

last few decades to model electrostatic interactions in complex molecular systems,

including the influence of polarization effects on structural refinement and the

analysis of structure/function relations. However, rigorous and practical

approaches to obtain ab initio quality electrostatic potentials have yet to be

established. Several techniques have emerged within the field of linear scaling

methods, based on the fragmentation of extended system into small molecular

domains and the subsequent calculation of the properties of the entire system from

accurate quantum chemistry calculations of the constituent fragments. The

recently developed MOD-QM/MM hybrid method belongs to this general class of

linear scaling approaches, and is based on a simple space domain decomposition

scheme and the iterative self-consistent treatment of the constituent molecular

fragments according to state-of-the-art QM/MM methods. The resulting

methodology complements previous approaches and extends the capabilities of

current QM/MM hybrid methods to the description of electrostatic interactions

that explicitly include mutual polarization effects in both the QM and MM layers.

Recent applications of the MOD-QM/MM methodology have been

focused on the development computational structural models of macromolecules

of biological interest, including photosystem II and guanine DNA quadruplexes,

demonstrating the capabilities of the method as applied to systems where

33

electrostatic interactions are critical. In addition to the analysis of MOD-QM/MM

structural models in terms of calculations of vibrational, electronic, and magnetic

properties and direct comparisons with FTIR, NMR, and EPR spectroscopy,

MOD-QM/MM structural models have been used as reference configurations in

conjunction with the development and application of post-QM/MM structural

refinement schemes. The post-QM/MM refinement methodology allowed for the

assessment of MOD-QM/MM molecular structures of similar energy (e.g., within

~5 kcal mol-1 from the minimum energy configuration) in terms of direct

comparisons with high-resolution spectroscopic data. The method has been

developed and implemented in terms of simulations of EXAFS spectra and the

minimization of 2χ -deviations between simulated and experimental spectra by

adjustment of nuclear coordinates, subject to the constraint of minimum

displacements relative to the reference MOD-QMM minimum energy

configuration. The same methodology could be implemented, more generally,

with other types of spectroscopic data (e.g., FTIR, NMR) whenever the

experimental data are available so long as the spectroscopic properties can be

readily computed. Such post-QM/MM refinement methods are particularly

valuable for the analysis of complex systems with rugged potential energy

landscapes, where many configurations have similar energy, including

metalloproteins for which the most accurate and reliable information on the

structure of prosthetic metal clusters comes from high-resolution spectroscopy.

The reviewed applications addressed fundamental questions that for many

years have remained largely elusive to rigorous first-principles examinations,

showing that QM/MM hybrid methods where polarization effects are explicitly

considered are powerful tools of modern computational chemistry when applied in

conjunction with high-resolution spectroscopy. It is, therefore, expected that these

QM/MM techniques will continue to make many more important contributions in

the characterization of structure and reactivity of biological macromolecules.

Acknowledgements V.S.B. acknowledges a generous allocation of DOE supercomputer time

from the National Energy Research Scientific Computing Center (NERSC) and financial support

from the Grants NSF ECCS # 0404191, NSF CHE # 0345984, DOE DE-FG02-07ER15909, NIH 2R01-

GM043278 and the US-Israel BSF R08164. G.W.B. acknowledges support from NIH GM32715.

34

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43

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44

H

MM

QM

H

MM

QM

Figure 1. QM/MM fragmentation of the system into a reduced QM

layer (region X) and the surrounding molecular environment

described as a MM layer (region Y).

45

QM layer (green) centered on domain R 1, MM layer (gray sticks)

QM layer (green) centered on domain R n, MM layer with updated

charges (balls & sticks). 1st cycle complete.

QM layer (green) centered on domain R 2, MM layer (gray sticks),

with updated charges on R1 (balls & sticks)

QM layer (green) centered on domain R n-1, MM layer (gray sticks),

with updated charges on all domains up to R n-1

(balls & sticks)

QM layer (green) centered on domain R 1, MM layer (gray sticks)

QM layer (green) centered on domain R n, MM layer with updated

charges (balls & sticks). 1st cycle complete.

QM layer (green) centered on domain R 2, MM layer (gray sticks),

with updated charges on R1 (balls & sticks)

QM layer (green) centered on domain R n-1, MM layer (gray sticks),

with updated charges on all domains up to R n-1

(balls & sticks)

Figure 2. Schematic representation of the MOD-QM/MM method in terms of an iterative cycle of

QM/MM calculations with QM layers (in green) defined on different molecular domains for

different steps of the cycle (Gascon et al. 2006a). Balls and sticks represent molecular domains with

ESP atomic charges updated in previous steps of the cycle.

46

64 quantum atoms

2000 classical atoms

64 quantum atoms

2000 classical atoms

Classical preoptimization

QM/MM optimization

QM/MM Set up

X-ray StructureAddition of water molecules

Sphere of 15 Å from the Mn cluster

Addition of water molecules

Sphere of 15 Å from the Mn cluster

QM/MM ModelQM/MM Model

A344

H332

D170

E333

E354

E189

D342

A344

H332

D170

E333

E354

E189

D342

A344

H332

D170

E333

E354

E189

D342

Figure 3. Left: Structural model of the OEC of PSII, including QM layer (balls and sicks) and MM layer

(wires). Right: QM/MM setup of the OEC of PSII.

47

Optimized coordinatesEXAFS spectrum at the

optimized coordinates

Optimized coordinatesEXAFS spectrum at the

optimized coordinates

X-ray diffraction P-EXAFSX-ray diffraction P-EXAFS

Initial structure Target functionInitial structure Target function

∑=i

i22 χχ ( )∑ −=

iii ExpCalc

N22

1

1χ

E=22χ

FEFF83FEFF83

New structure New P-EXAFSNew structure New P-EXAFS

< ε ?2χ < ε ?2χNo

Yes

Ab initio EXAFS (Multiple scattering theory: Green function formalism)

Optimization algorithm

External procedure

Figure 4. Post-QM/MM structural-refinement protocol based on simulations of X-ray absorption spectra with

FEFF8 and direct comparisons with experimental data (Sproviero et al. 2008c).

48

Polarized EXAFS: Provide information on 3D structur e

EXAFS intensity

Polarized beamOriented single crystals

( )θεε cosr=⋅r)2if r⋅∝ ε)

hν

M

Lrε)hν

M

Lrε)

M

Lε) θθθθr

M

Lε) θθθθr

EXAFS amplitude is orientation dependent.

if then

Contribution of L to the EXAFS is optimized The L contribution is now minimized

L

Mhν

ML

M-L (r) vector parallel to ε) M-L vector perpendicular to ε)

hνrr

ε) ε)

Contribution of L to the EXAFS is optimized The L contribution is now minimized

L

Mhν

ML

M-L (r) vector parallel to ε)M-L (r) vector parallel to ε) M-L vector perpendicular to ε)M-L vector perpendicular to ε)

hνrr

ε) ε)

Figure 5. Schematic representation of the process of polarized X-ray absorption as a function of

the angle between the polarized vector and the metal-ligand orientation in single crystals.

49

0 1 2 3 401234567

0 1 2 3 401234567

0 1 2 3 401234567

Reduced Distance [Å]

FT

Mag

nitu

de a b c

ExperimentalQM/MM

Mn-MnMn-OMn-MnMn-ca

0 1 2 3 401234567

0 1 2 3 401234567

0 1 2 3 401234567

Reduced Distance [Å]

FT

Mag

nitu

de a b c

ExperimentalR-QM/MM

Mn-MnMn-O

Mn-MnMn-ca

0 1 2 3 401234567

0 1 2 3 401234567

0 1 2 3 401234567

Reduced Distance [Å]

FT

Mag

nitu

de a b c

ExperimentalQM/MM

Mn-MnMn-OMn-MnMn-ca

0 1 2 3 401234567

0 1 2 3 401234567

0 1 2 3 401234567

Reduced Distance [Å]

FT

Mag

nitu

de a b c

E x p e r i m e n t a l

R - Q M / M M

Mn-MnMn-O

Mn-MnMn-ca

Figure 6. Comparison of the experimental polarized EXAFS data (red) (Yano et al. 2006) and the

corresponding calculated spectra (blue) obtained with the MOD-QM/MM (top) and R-QM/MM (bottom)

models (Sproviero et al. 2008c).

50

0 1 2 3 40

2

4

6

8

10

ExperimentalR-QM/MM

Mn-MnMn-O

Mn-MnMn-ca

0 1 2 3 40

2

4

6

8

10

ExperimentalQM/MM

Mn-MnMn-O

Mn-MnMn-ca

4 5 6 7 8 9 10 11 12

-8

-4

0

4

8

4 5 6 7 8 9 10 11

-8

-4

0

4

8

k [Å-1]

EX

AF

S χ

k3

Reduced Distance [Å]

FT

Mag

nitu

de

0 1 2 3 40

2

4

6

8

10

ExperimentalR-QM/MMExperimentalR-QM/MM

Mn-MnMn-O

Mn-MnMn-ca

0 1 2 3 40

2

4

6

8

10

ExperimentalQM/MM

Mn-MnMn-O

Mn-MnMn-ca

0 1 2 3 40

2

4

6

8

10

ExperimentalQM/MMExperimentalQM/MM

Mn-MnMn-O

Mn-MnMn-ca

4 5 6 7 8 9 10 11 12

-8

-4

0

4

8

4 5 6 7 8 9 10 11

-8

-4

0

4

8

4 5 6 7 8 9 10 11 12

-8

-4

0

4

8

4 5 6 7 8 9 10 11

-8

-4

0

4

8

k [Å-1]

EX

AF

S χ

k3

Reduced Distance [Å]

FT

Mag

nitu

de

Figure 7. Comparison of the experimental (red) (Dau et al. 2004) isotropic EXAFS spectrum in k-space

(left) and FT EXAFS magnitude (right) and the calculated spectra (blue) obtained with the QM/MM (top)

and R-QM/MM (bottom) models (Sproviero et al. 2008c).

51

D170

E333

H332E189

A344

D342

CP43-E354

Mn(1)

Mn(2)

Mn(3)

Mn(4)

Ca

D170

E333

H332E189

A344

D342

CP43-E354

Mn(1)

Mn(2)

Mn(3)

Mn(4)

Ca

MOD-QM/MM (Green)R-QM/MM (Colored)D170

E333

H332E189

A344

D342

CP43-E354

Mn(1)

Mn(2)

Mn(3)

Mn(4)

Ca

D170

E333

H332E189

A344

D342

CP43-E354

Mn(1)

Mn(2)

Mn(3)

Mn(4)

Ca

MOD-QM/MM (Green)R-QM/MM (Colored)

Figure 8. Superposition of the refined (R)-QM/MM (colored) and MOD-QM/MM (green) reference

structure of the OEC of PSII (Sproviero et al. 2008c). The R-QM/MM model reproduces the

experimental isotropic (Dau et al. 2004) and polarized (Yano et al. 2006) EXAFS spectra and is

consistent with XRD models (Ferreira et al. 2004; Loll et al. 2005).

52

Mn(2)

Mn(1)

Mn(1)

Mn(2)

Mn(3)

Mn(2)

Mn(1)

(a) (b)

(c)

Figure 9. Molecular structures of (a) complex 1, [MnIVMnIII (m-O)2-(H2O)2(terpy)2]3+ (terpy =

2,2’:6,2’’-terpyridine); (b) complex 2, [MnIIIMnIV(m-O)2(phen)2]3+ (phen = 1,10-

phenanthroline) ; and (c) complex 3, [Mn3O4(bpy)4(OH2)2]4+ (bpy = 2,2’-bipyridine)

(Sproviero et al. 2006a). Color key: red = oxygen, blue = nitrogen, gray = carbon, white =

hydrogen, and purple = manganese.

53

Figure 10. DFT–QM/MM structural model of the OEC of PSII

(Sproviero et al. 2006b). Fast- and slow-exchanging substrate water

molecules, Wfast and Wslow, are coordinated to Mn(4) and Ca2+,

respectively (Sproviero et al. 2008e). Red (blue) indicate an increase

(decrease) in ESP atomic charges due to S1/S2 oxidation. Bright colors

indicate changes of approximately 15–20%. All amino acid residues

correspond to the D1 protein subunit unless otherwise indicated.

54

Wf

W*

Ws

Mn(1)

Mn(2)

Mn(3)

Mn(4)Cl−−−−

Ca2+

Y161D170

A344

E333

H332E189

W1*

W2*

S1: 4.14 ÅS2: 3.87 Å

S1: 2.23 ÅS2: 2.09 Å

S1: 2.42 ÅS2: 2.40 Å

S1: 3.28 ÅS2: 4.47 Å

S1: 2.42 ÅS2: 3.42 Å

Figure 11. Structural model of the OEC of PSII in the S1 and S2 states. Ligated and exchanging water molecules are

labeled Ws (Wslow ), Wf (Wfast ) and W*, respectively. All amino acid residues correspond to the D1 protein subunit,

unless otherwise indicated (Sproviero et al. 2008e).

55

Figure 12. Structural model of the OEC of PSII in the S1 and S2 states at

the transition state configurations for exchange of the fast exchanging water

from Mn(4). The water-exchange energy barriers in the S1 and S2 states are

7.9 and 8.4 kcal mol/K, respectively. Substrate and exchanging water

molecules are labeled Wf (Wfast) and W*, respectively. All amino acid

residues correspond to the D1 protein subunit, unless otherwise indicated

(Sproviero et al. 2008e).

56

WfWs

W1*

W2*

W*

Mn(1)

Mn(2)

Mn(3)

Mn(4)Cl 0000

Ca2+

Y161D170

A344 E333

H332E189

3.38 Å3.33 Å

Figure 13. Structural model of the OEC of PSII in the S1 and S2 states at the

transition state configurations for water exchange from Ca2+. The water-exchange

energy barriers in the S1 and S2 states are 19.3 and 16.6 kcal mol-1, respectively.

Substrate and exchanging water molecules are labeled Ws (Wslow) and W1*,

respectively. All amino acid residues correspond to the D1 protein subunit, unless

otherwise indicated (Sproviero et al. 2008e).

57

M+

M+

M+

M+

M+T5

T6 T7 T8

T8

T7

T5

T6

G12G1

G4

G9

G10

G11

G3

G12

G4G1

G2G10

G2G3

G11

G9

5’

3’

3’

5’

M+M+

M+M+

M+M+M+

M+M+

M+M+M+T5

T6 T7 T8

T8

T7

T5

T6

G12G1

G4

G9

G10

G11

G3

G12

G4G1

G2G10

G2G3

G11

G9

5’

3’

3’

5’

Figure 14. Schematic diagram of the DNA quadruplex of (G4)

of Oxytricha Nova. Guanines are shown as rectangles in each

planar quartet. Thymines (T5–T8) are shown at dashes in the

loops and the monovalent cations (M+) as spheres in between the

planes (Newcomer et al. 2009).

58

N

N

N

N

O

R

H

N

H

H

H

N

N

N

N

O

R

H

N

H

H

H

N

N

NN

O

R

H

N H

HH

N

N

NN

O

R

H

NH

HH

Aromatic H

Imino H

N

N

N

N

O

R

H

N

H

H

H

N

N

N

N

O

R

H

N

H

H

H

N

N

NN

O

R

H

N H

HH

N

N

NN

O

R

H

NH

HH

N

N

N

N

O

R

H

N

H

H

H

N

N

N

N

O

R

H

N

H

H

H

N

N

NN

O

R

H

N H

HH

N

N

NN

O

R

H

NH

HH

Aromatic H

Imino H

Figure 15. Imino (red circles) and aromatic (blue circles) protons as

defined in each planar quartet (Newcomer et al. 2009).

59

Figure 16. Correlation between experimental 1H NMR and calculated chemical shifts

for DNA quadruplex models (Newcomer et al. 2009). The models were obtained by

structural refinement based on the MOD-QM/MM method (black dots), including an

explicit description of polarization effects, and by using the distribution of atomic

charges defined by the Amber MM force field (red triangles).

60

Figure 7. Each panel shows one molecular domain of 4

Figure 17. The fragmentation of the DNA quadruplex partitions each quartet into four

molecular domains (R1–R4), highlighted by dashed lines. During each step of the MOD-

QM/MM cycle, only the guanine moiety with hydrogen atoms involved in hydrogen

bonds within the QM layer is allowed to relax while the configuration of the other moiety

is fixed as determined by previous optimization steps in the cycle (Newcomer et al. 2009).

61

Table 1. Oxidation numbers, Mulliken spin population analyses (in parenthesis) and ab initio exchange

coupling constants, Jij , computed at the DFT B3LYP level with the TZV basis set for the high-valent

multinuclear oxomanganese complexes 1, 2 and 3 introduced in the text (see Figure 9).

Complex Mn(1) Mn(2) Mn(3) J12 (cm-1) J23 (cm-1)

1 III(+3.8) IV(-2.6) - 298(300)* -

2 III(+3.9) IV(-2.6) - 295(268-300)* -

3 IV(-1.2) IV(-1.2) IV(+2.8) 211(182)* 70(98)*

*Values in parenthesis indicate experimental values of coupling constants(Cooper and

Calvin 1977; Cooper et al. 1978).