Computational biochemistry ferenc Bogár György Ferency
π-π , π-HN, π-HO stackings
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- Chapter 2. Molecular Mechanics
- What is it used for
- What is needed
- 1. Introduction
- 2. Traditional Molecular Mechanics Methods
π-π , π-HN, π-HO stackings have interaction between aromatic electron systems and C-H, O-H, N-H atoms. These interactions are dipole, quadrupole etc. interactions. H-bonds X-H…Y, where X and Y as pilar atoms have greater electronegativity. The energy stability is increasing and the geometries are deformed. F−H…:F (161.5 kJ/mol), O−H…:N (29 kJ/mol), O−H…:O (21 kJ/mol), N−H…:N (13 kJ/mol), N−H…:O (8 kJ/mol), HO−H…:OH (18 kJ/mol). X−H…Y system: X−H distance is typically ca. 110 pm, whereas H…Y distance is ca.160 to 200 pm. Databases, The 3D structures (XRD, NMR) of biomolecules are deposited in http://www.pdb.org, The known sequeces of the living systems are summerized in http://www.expasy.org/. Chapter 2. Molecular Mechanics (Tamás Körtvélyesi)
like) ligands to target(s). What is needed? The knowledge of the structure of biomolecules, intra- and intermolecular interactions to stabilize their structure (covalent, polar and weakly polar interactions) are fundamental to understand this chapter. The basic knowledge in physics and physical chemistry is also important. 1. Introduction In the early forties of the last century Westheimer, an organic chemist suggested a molecular model: the atom sin the molecule is connected by springs. The structure can be calculated considering the force constants of the springs between the atoms and the non-bonded interactions. The idea is simple but no computers were available at that time. Only in the fifties-sixties were developed algorithms which use of the idea mentioned above. The method is simple and fast to find the conformations, intra- and intermolecular interactions, electrostatic properties of small molecules and large molecules, too. Molecular mechanics is the only method to handle large (bio)molecular systems with 100-200 thousands of atoms [1]. The algorithm makes possible to apply in computer assisted drug design. 2. Traditional Molecular Mechanics Methods On a multidimensional Born-Oppenheimer surface the nuclear positions are given by the function described in an Eq. 2.1:
The potential energy function in a molecule can be partitioned by the deformations Eq. 2.2
where Vstretching is potential of the deformation in the bonds, Vbending is the potential of the deformation in the angle bending and Vtorsion is the potential of the deformation in the torsion angle (covalent deformation), Voop is the out-of-plane deformation, Vcross is the cross functions of the covalent interactions . See Figure 2.1. Vnb is the potential energy of the non-bonding (the Coulomb and the van der Waals) interactions [1]. This sum is called steric energy. 
Figure 2.1. Deformations in molecular mechanics handled by the all-atom and united atom models (see later). Covalent bonded interactions The most simple function to describe the bond deformations is given by the Hooke’s law Eq. 2.3.
This function (which is suitable for small deformations, a harmonic vibrational potential) with the parameters of ki,stretching (stretching force constants) and ri,o, natural bond length (the bond distances in an ideal strain free bond) for the individual bonds. E.g. ki,stretching is 272 kJ/mol. rCsp3-Csp3,o is 1.54 Ǻ. The best ri,o values can be obtained by electron diffraction. At X-ray diffraction, the bond length of C-H has to correct by 0.015 Ǻ because of the electronegativity difference between C and H. The Hooke’s law is valid for small deformations, at strained systems the function is not precise. The equation can be modified as Eq. 2.4
kstretching, k’stretching , k’’stretching are the force constants. In a strained molecular system the Morse function (the potential of the bond deformation in singlet state with dissociation) can be used Eq. 2.5 (see Figure 2.2).
De.iis the dissociation constant of the ith bond. αi, and Xi are the Morse constants of the bonds and the deformation of the bonds (Xi = r – ri,o), respectively. The Morse constant of a bond pair can be calculated by
where ke,i is the force constant of the ith bond. 
Figure 2.2. Dissociation energy profiles: harmonic and the energy profiles by Morse function. De is the dissociation energy, D0is the ZPVE (zero point vibrational energy, D0 = De + ZPVE, v= 0) corrected dissociation energy, re is the equilibrium distance, v is the energy level The simplest function of the deformation in the angle can be described by Eq. 2.7
ki,bending and θi,o are the bending force constants and the natural angle of three atoms in an ideal strain free bonds, respectively. This equation is valid for ca. 10 degrees deformation. At larger deformation in a strained molecule a cubic term can correct the potential Eq. 2.8
ki,bending, k’i,bending , k’’i,bending are the bending force constants. Considering a rotation around a bond by a 0 to 180 degree, we can find minima and maxima (see the rotational energy profile of ethane, Figure 2.3.). In ethane molecule the staggered conformations are connected by eclipsed conformations. 
Figure 2.3. Rotational energy profile in ethane (Potential energy vs. dihedral angle of H-C-C-H) The torsion energies can be described by a Fourier series of terms in Eq. 2.9
ω is the dihedral angle, γ is the phase factor (torsion angle at minimum). n is the number of different rotational positions. At sp 3 carbon atoms (C-C-C-C) n = 3, γ = 0 ◦ , at sp2◦ (C-C=C-C), n = 2, γ = 180 ◦. In AMBER FF three torsion expression is used
The three terms have different meanings: (i) the first expression is the description of the dipole-dipole, van der Waals and other interactions between atoms, (ii) the second expression is on the conjugation and/or hyperconjugation, (iii) the third expression is on the steric effects between 1,4 atoms. 
Figure 2.4. Molecular mechanics handled by all-atom and united atom models The molecules can be handled by all-atom and united atom model (see Figure 2.4.). It means that the models consist of the all-atoms in the molecules or only the heavy atoms (not H-atoms) and the H-atoms connected to polar atoms (not C-atoms, but N-, O-, S-, P-atoms with higher electronegativity than C-atoms). The charges, masses and force constants are corrected to the all-atom parameters. It is important to decrease the degrees of freedom for the geometry optimization (see Chapter 3). The improper torsions and out-of–plane bending motion (see Figure 2.5.) is for stabilizing ring structures and the chirality of the structures (e.g. at the united atom model without out-of-plane restriction the chirality of the Cα can be changed). 
Figure 2.5. Out-of-plane bending with θ angle The Voop potential can be handled by a quadratic equation Eq. 2.11.
The Voop potential is given by Eq. 2.12, where ω is the torsion angle starting in the centre.
In class 1, 2 and 3 force fields cross terms are applied: stretch-stretch, stretch-bend and bend-bend
In the Urey-Bradley force field the 1,3-nonbonded interactions are handled explicitly.
Δρi is the change in the distance between non-bonded atoms. Ki, Li, Mi, ki, li and mi are the parameters of the force field.
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