Linked-atom treatment

Figure 12-1. Illustration of the difference between the pseudobond approach and the conventional link atom approach in the treatment of the QM/MM boundary problem...

In an attempt to aid interpretation of the IR spectrum of MbCO we decided to model the full protein by use of a hybrid quantum mechanics/molecular mechanics approach (QM/MM), to evaluate changes in the CO stretching frequency for different protein conformations. The QM/MM method used [44] combines a first-principles description of the active center with a force-field treatment (using the CHARMM force field) of the rest of the protein. The QM-MM boundary is modeled by use of link atoms (four in the heme vinyl and propionate substituents and one on the His64 residue). Our QM region will include the CO ligand, the porphyrin, and the axial imidazole (Fig. 3.13). The vinyl and propionate porphyrin substituents were not included, because we had previously found they did not affect the properties of the Fe-ligand bonds (Section 3.3.1). It was, on the other hand, crucial to include the imidazole of the proximal His (directly bonded to the... 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

I. Antes, W. Thiel, On the Treatment of Link Atoms in Hybrid Methods, in Combined Quantum Mechanical and Molecular Mechanical Methods, J. Gao, M. A. Thompson (eds), American Chemical Society, Washington, DC, 1998, pp. 50-65. 

The construction in Fig. 6 may at first sight seem like that used in QM/MM approaches. However, the metal is the only QM atom plus there are no link atoms. The LFMM thus provides a uniform, seamless theoretical treatment of the entire molecule. 

In contrast, LFMM/LFMD is much faster and provides a uniform treatment over the whole molecule (i.e., there are no link atoms). Moreover, proteins behave like giant ligands. They may be exquisitely complex and varied but, at the end of the day, the intrinsic bonding interaction between, say, Cu(II) and the N of imidazole is essentially the same whether the imidazole is a free ligand or happens to be part of a histidine which is, in turn, connected to a peptide backbone. Hence, if an LFMM FF can be constructed for small coordination complexes containing biologically relevant donors, then it should work for whole proteins. 

The double bonds of the rubber molecules all have the cis, or Z, configuration. This causes bends in the chains that make the molecules less able to crystallize. To prevent the molecules from slipping past each other when the rubber is stretched, the molecules are cross-linked by treatment with sulfur in a process called vulcanization. Although the exact details of vulcanization are not known, links between different chains are formed by one or two sulfur atoms ... 

The link atom did not interact with the MM atoms, either electrostatically or through van der Waals terms. The treatment of the interactions of link atoms has been explored by Reuter et al. [145] and Antes and Thiel [147,150]. It has been found to be inadvisable to exclude link atoms from interactions with the MM charges. In fact, it is preferable for all QM atoms to interact with the same set of MM atoms, which has implications also for the definition of any... 

The evolution of HBr in the bromination reactions and the uptake of one bromine atom per ring indicate substitution at a secondary allylic carbon atom. The ease of oxidation and cross linking of the polymers and the presence of hydroxyl groups imply the intermediate formation of hydroperoxides on allylic carbon atoms. Treatment of the hydroxyl-containing polymer with benzoyl chloride indicates that the bulk of the hydroxyl groups are on secondary carbon atoms, since tertiary hydroxyl groups would tend to be replaced by chlorine. Although these results do not permit the elimination of structure B, it appears that the bulk of the structural units in the polycyclopentadiene corresponds to 1,2- addition (A). 

The only difference with the case of solutions arises from the treatment of the electrostatic interaction between the link atom and the classical atoms. In a first version implemented in the CHARMM package [10], this interaction was simply neglected (the QQ link). An alternative version in which the link atom interacts with all the atoms has been added (the HQ link). Due to the size of the reactive parts of... 

Several related methods have been developed during the past years. The Achilles heel of this approach is the treatment of the interaction between the link atom and classical part of the system, and the proper simulation of the bonds between the two kinds of subsystems [17]. Other shortcomings come from the nature of the link atom. 

Antes 1, Thiel W. On the treatment of link atoms in hybrid methods. In Gao J, Thompson MA, editors. Hybrid Quantum Mechanical and Molecular Mechamcal Methods Proceedings of ACS Symposium Series. Washington, DC ACS 1998. 712 50-65. 

In a semiempirical framework, the inclusion of extra hydrogen atoms can be avoided through the use of the adjusted connection atom (ACA) approach, in which a boundary atom is parameterized to mimic a methyl group (66). Alternative approaches do not involve addition or change of atom types, but a special treatment of the electronic orbitals is employed for the QM/MM boundary atoms. In the local frozen orbital approach (49,67), one singly occupied orbital is defined to be along the truncated bond that is neglected (i.e., frozen) in the wave function calculation. It has been found that if carried out carefully, the local orbital treatment and the link atom approach are of similar accuracy (68). 

In the QM/MM version introduced by Karplus et al. semiempirical methods of MNDO and AMI type in combination with MM treatment were suggested to determine the APE surface for further MD simulation of the system as a whole. As in the works above, the authors discuss in detail the inter-region QM-MM interaction problem and the saturation of the free valences by link atoms when the QM/MM separation cuts a bond. The method is implemented in a program package called CHARMM . Among... 

The results obtained are in line with former studies [10] that compared the link atom and the LSCF approaches at soniempirical level and concluded that they give comparable results when applied cautiously. However, the LSCF approach is conceptually more appealing and its current Huzinaga equation-based implementation allows for the first time to perform calculations at the ab initio level without orlhogonalizing the basis set to the frozen orbitals. Further developments, such as gradient calculations, inclusion of electron correlation effects from the QM side and the treatment of periodic boundary conditions and to perform molecular dynanfics simulations on the MM side will be performed. These developments will allow a more thorough exploration of the optimal parameters of the HLSCF method with the aim of providing a versatile QM/MM tool to the community. 

Of the three types of boundary treatment, the link atom method is the simplest both conceptually and in practice, and is hence the most widely used. The boundary atom and in particular the frozen orbital methods can potentially achieve higher accuracy but require careful a priori parametrization and bear limitations on transferability (Senn and Thiel 2009). 

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