11.5.3 Extension to Macromolecules: Fragmented EFP Scheme

Macromolecules such as proteins or DNA present a large number of electronic structure problems (photochemistry, redox chemistry, reactivity) that can be described within QM/EFP framework. EFP has been extended to deal with such complex systems via the so-called fragmented EFP scheme (fEFP). The current Q-Chem implementation allows one to (i) compute interaction energy between a ligand and a macromolecule (both represented by EFP) and (ii) to calculate the excitation energies, ionization potentials, electronic affinities of a QM moiety interacting with a fEFP macromolecule using QM/EFP scheme (see Section 11.5.2). In the present implementation, the ligand cannot be covalently bound to the macromolecule.

There are multiple ways to cut a large molecule into units depending on the position of the cut between two covalently bound residues. An obvious way to cut a protein is to cut through peptide bonds such that each fragment represents one amino acid. Alternatively, one can cut bonds between two atoms of the same nature (carbonyl and carbon-$\alpha$ or carbon-$\alpha$ and the first carbon of the side chain). The user can choose the most appropriate way to cut.

Consider a protein ($P$) consisting of $N$ amino acids, $A_1{A}_2\mathrm{\dots }{A}_N$, and is split into $N$ fragments ($A_i$). The fragments can be saturated by either Hydrogen Link Atom⁸⁸¹ (HLA) or by mono-valent groups of atoms from the neighboring fragment(s), called Cap Link Atom (CLA) hereafter. If fragments are capped using the HLA scheme, the hydrogen is located along the peptide bond axis and at the distance corresponding to the equilibrium bond length of a CH bond:

$$P=A_{1}H+\sum _{i=2}^{N-1}H{A}_{i}H+H{A}_N$$

(11.69)

In the CLA scheme, the cap has exactly the same geometry as the respective neighboring group. If the cuts are made through peptide bonds (one fragment is one amino acid), the caps ($C^i$) are either an aldehyde to saturate the -N(H) end of the fragment, or an amine to saturate the -C(=O) extremity of the fragment.

$$P=A_1{C}^2+\sum _{i=2}^{N-1}{C}^{i-1}{A}_i{C}^{i+1}+C^{N-1}{A}_N$$

(11.70)

Q-Chem provides a two-step script, prefefp.pl, located in $QC/bin which takes a PDB file and breaks it into capped fragments in the gamess format, such that the EFP parameters for these capped fragments can be generated, as explained in Section 11.5.7. As the EFP parameters are generated for each capped fragment, the neighboring fragments have duplicated parameter points (overlapping areas) in both the HLA and CLA schemes due to the overlapping caps. Since multipole expansion points and polarizability expansion points are computed on each capped residue by the standard procedure, the multipole (and damping terms) and polarizabilities need to be removed ($C^{\mathrm{\varnothing }}$) from the overlapping areas.

Equations (11.69) and (11.70) become:

$$P=A_1{C}^{\mathrm{\varnothing }}+\sum _{i=2}^{N-1}{C}^{\mathrm{\varnothing }}{A}_i{C}^{\mathrm{\varnothing }}+C^{\mathrm{\varnothing }}{A}_N$$

(11.71)

The details concerning this removing procedure are presented in Section 11.5.7.

Once these duplicate parameters are removed from the EFP parameters of the capped fragments, the EFP-EFP and QM-EFP calculations can be conducted as usual.

Currently, fEFP includes electrostatic and polarization contributions, which appear in EFP(ligand)/fEFP(macromolecule) and in QM/fEFP calculations (note that the QM part is not covalently bound to the macromolecule). Consequently, the total interaction energy ($E^{\mathrm{tot}}$) between a ligand ($L$) and a protein ($P$) divided into fragments is:

$$E^{\mathrm{tot}}(P-L)=E^{\mathrm{elec}}(P-L)+E^{\mathrm{pol}}(P-L)$$

(11.72)

The electrostatics is an additive term; its contribution to fragment-fragment and ligand-fragment interaction is computed as follows:

$$E^{\mathrm{elec}}(P-L)=\sum _i^N{E}^{\mathrm{elec}}\left(C^{\mathrm{\varnothing }}{A}_{i=1}{C}^{\mathrm{\varnothing }}-L\right)$$

(11.73)

The polarization contribution in an EFP system (no QM) is:

$$E^{\mathrm{pol}}(P-L)=-\frac{1}{2}\sum _{k\in P,L}{\mu }^k{F}^{\mathrm{mult},k}+\frac{1}{2}\sum _{k\in P}{\mu }^k{F}^{\mathrm{mult},k}$$

(11.74)

The first term is the polarization energy obtained upon convergence of the induced dipoles of the ligand (${\mu }_{\mathrm{efp}}^k(L)$) and all fragments (${\mu }_{\mathrm{fefp}}^k(A_i)$). The system is thus fully polarized, all fragments (A${}_i$ or L) are polarizing each other until self-consistency.

	${\mu }_{\mathrm{efp}}^k(L)$	$={\displaystyle \sum _{k\in A_i}}{\alpha }^k(F^{\mathrm{mult},k}+F^{\mathrm{ind},k})$		(11.75)
	${\mu }_{fefp}^k(A_i)$	$={\displaystyle \sum _{j\ne i}}{\displaystyle \sum _{k\in L,A_j}}{\alpha }^k(F^{\mathrm{mult},k}+F^{\mathrm{ind},k})$		(11.76)

The second term of Eq. (11.74) is the polarization of the protein by itself; this value has to be subtracted once the induced dipoles (Eq. 11.75) converged.

The LA scheme is available to perform QM/fEFP job. In this situation the fEFP has to include a macromolecule (covalent bond between fragments). This scheme is not able yet to perform QM/fEFP/EFP in which a macromolecule and solvent molecules would be described at the EFP level of theory.

In addition to the HLA and CLA schemes, Q-Chem also features Molecular Fragmentation with Conjugated Caps approach (MFCC) which avoids the issue of overlapping of saturated fragments and was developed in 2003 by Zhang.^{1096, 1095} The MFCC procedure consists of a summation over the interactions between a ligand and capped residues (CLA scheme) and a subtraction over the interactions of merged caps ($C^{i+1}{C}^{i-1}$), the so-called “concaps”, with the ligand. $N-1$ concap fragments are actually used to subtract the overlapping effect.

$$P=A_1{C}^2+\sum _{i=2}^{N-1}{C}^{i-1}{A}_i{C}^{i+1}+C^{N-1}{A}_N-\sum _i^{N-1}{C}^{i+1}{C}^{i-1}$$

(11.77)

In this scheme the contributions due to overlapping caps simply cancel out and the EFP parameters do not need any modifications, in contrast to the HLA or CLA procedures. However, the number of parameters that need to be generated is larger ($N$ capped fragments + $N-1$ concaps).

The MFCC electrostatic interaction energy is given as the sum of the interaction energy between each capped fragment ($C^{i-1}{A}_i{C}^{i+1}$) and the ligand minus the interaction energy between each concap ($C^{i-1}{C}^{i+1}$) and the ligand:

$$E^{\mathrm{elec}}(P-L)=\sum _i^N{E}^{\mathrm{elec}}\left(C^{i-1}{A}_i{C}^{i+1}-L\right)-\sum _i^{N-1}{E}^{\mathrm{elec}}\left(C^{i-1}{C}^{i+1}-L\right)$$

(11.78)

The main advantage of MFCC is that the multipole expansion obtained on each capped residue or concap are kept during the $E^{\mathrm{elec}}(P-L)$ calculation. In the present implementation, there are no polarization contributions. The MFCC scheme is not yet available for QM/fEFP.